LIBRARY 

OF  THK 

UNIVERSITY  OF  CALIFORNIA. 

©IFT    OF 

BETA  THBTA  Pi 

Class 


\ 


PHYSICAL 


LABORATORY  MANUAL 


FOB  SECONDARY  SCHOOLS 


BY 


S.    E.   COLEMAN,  S.B.,  A.M. 

HEAD   OF  THE   SCIENCE   DEPARTMENT,   AND   TEACHER  OF  PHYSICS 
IN    THE   OAKLAND   HIGH   SCHOOL 


•FTHE 

{    UNIVERSITY  ) 


NEW  YORK  .:•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK   COMPANY 


COPYRIGHT,  1903,  BY 
8.  E.  COLEMAN. 

ENTERED  AT  STATIONERS'  HALL,  LONDON. 
COLEMAN.    PHY.  LAB.  MAN. 

W.    P.    2 


OF  THE 

UNIVERSITY 

OF 


PREFACE 

THE  number  and  variety  of  laboratory  manuals  in  physics 
now  on  the  market  are  so  large  that  the  merits  of  a  new  one 
are  hardly  to  be  sought  in  novelty  of  content,  but  rather  in 
the  utility  of  the  material  chosen  from  the  abundant  common 
store  and  in  the  method  of  presenting  it.  Believing  that  these 
are  both  matters  of  great  importance  and  that  the  possibilities 
of  improvement  in  them  have  not  yet  been  exhausted,  the 
author  has  written  this  manual  in  the  hope  that  it  will  con- 
tribute toward  this  end. 

Since  both  the  choice  of  material  and  the  method  of  treat- 
ment are  in  a  large  measure  determined  by  the  view  enter- 
tained in  regard  to  the  place  and  function  of  the  laboratory 
work  in  the  course  in  physics,  a  brief  statement  on  this  point 
seems  desirable.  Rejecting  the  extreme  view  that  little  im- 
portance is  to  be  attached  to  any  part  of  the  work  except  the 
laboratory  course,  as  well  as  the  opposite  extreme  which  rele- 
gates this  part  of  the  work  to  the  position  of  a  supplementary 
adjunct  to  the  old  form  of  text-book  instruction,  the  author 
has  adopted  in  his  teaching  and  has  assumed  as  the  controlling 
principle  in  the  preparation  of  this  manual  the  view  that  the 
laboratory  course  should  stand  in  coordinate  relationship  to 
the  work  of  the  class  room;  which,  in  his  opinion,  should 
include  as  important  elements  qualitative  experimental  work 
by  the  teacher,  the  systematic  study  of  a  good  text-book,  as 
large  a  use  of  reference  books  as  time  and  opportunity  permit, 
a  constant  appeal  to  the  everyday  experience  of  the  pupils, 
and,  finally,  the  recitation  or  quiz,  in  which  the  information 
gleaned  from  the  several  sources  is  classified,  organized  into 
scientific  knowledge,  and  assimilated. 

This  point  of  view  presupposes  that  the  laboratory  work 

186375 


4  PREFACE 

will,  iu  general,  precede  the  recitation,  but  may  itself  be  pre- 
ceded by  experimental  work  by  the  teacher,  presenting  funda- 
mental phenomena  as  an  introduction  to  either  the  qualitative 
or  quantitative  work  of  the  laboratory.  It  is  also  assumed  that 
the  text-books  and  reference  books  are  legitimate  aids  toward 
the  interpretation  of  the  experiment  while  it  is  being  performed, 
and  that  the  reading  of  the  text  on  the  subject  of  the  experi- 
ment before  the  laboratory  hour  will  economize  time  in  the 
laboratory  and  lead  to  the  best  results. 

The  choice  of  experiments  has  been  governed  by  the  fol- 
lowing considerations :  — 

(1)  The  content  of  the  laboratory  course  should  be  rich  and 
varied,  and  the  manipulation  the  simplest  that  will  serve  the 
purpose.     Skill  in  manipulation  should  not  be  sought  for  its 
own  sake,  but  as  a  means  to  an  end,  the  end  being  a  satisfac- 
tory presentation  'of  physical  facts  and  principles.     Measure- 
ments with  vernier  and  micrometer  calipers,  the  diagonal  scale, 
the  spherometer,  etc.,  are  omitted,  since   the   knowledge  of 
their  use  finds  no  important  application  in  elementary  physics. 

(2)  There  are  many  qualitative  experiments  —  such  as  the 
reflection  or  refraction  of  a  beam  of  sunlight  in  a  darkened 
room  —  that  can  be  fully  appreciated  at  a  distance,  and  for 
which  the  recitation  room  affords  better  facilities  than  the 
laboratory.     Such  experiments  are  not  included  in  the  manual, 
which  is  limited  to  experiments  best  adapted  to  the  laboratory. 
But  the  course  includes  many  valuable  qualitative  experiments 
that  can  readily  be  provided  for ;  and  in  a  considerable  num- 
ber of  cases  they  present  phenomena  to  the  student  much  more 
satisfactorily   than    the   equivalent    class-room    experiments; 
although,  for  nearly  all  of  them,  the  latter  may  be  substituted 
if  the  teacher  prefers. 

(3)  In  order  that  there  might  be  opportunity  for  choice,  the 
number  of  experiments  has  been  made  considerably  greater 
than  most  teachers  will  require  of  their  classes.     The  alterna- 
tive experiments  serve  the  same  purpose,   and  increase  the 


PREFACE  5 

adaptability  of  the  manual  to  the  varying  equipment  of 
different  laboratories. 

The  exercises  marked  with  an  asterisk  in  the  Table  of  Con- 
tents, —  exclusive  of  Part  III  of  Exercise  10  and  Part  II  of 
Exercises  27,  34,  and  76,  —  are  suggested  as  a  minimum  course. 
They  cover  all  the  essentials  of  a  good  laboratory  course,  yet, 
for  the  most  part,  require  only  inexpensive  apparatus. 

The  method  of  treatment  presents  the  following  character- 
istic features :  — 

(1)  The  presentation  is  in  the  main  inductive,  but  not  ex- 
clusively so.     The  laboratory  course  is  regarded  as  an  impor- 
tant source  of  information  at  first  hand,  which  is  to  serve  as  a 
representative  (though  generally  incomplete)  basis  upon  which 
to  establish  the  theory,  rather  than  as  serving  merely  to  illus- 
trate or  exemplify  the  theory  dogmatically  presented  in  the 
text-book  and  the  recitation.     No  pains  have  been  spared  in 
the  effort  to  cast  the  experiments  in  such  form  that  the  pupil 
will  be  led  by  correct  reasoning  from  the  observed  facts  to  the 
legitimate  inferences.     Where  the  interpretation  of  results  and 
the  theory  of  experimental  methods  can  best  be  arrived  at 
deductively  from  established  theory,  this  method  is  employed. 

(2)  A   persistent   effort   is   made   throughout   to   stimulate 
thought  and  to  minimize  unreasoning,  mechanical  work.     To 
this  end  the  pupil  is  generally  required  to  arrive  at  his  results 
by  simple  analytical  processes,  which  to  be  employed  must 
be  understood,  and  the  understanding  of  which  involves  the 
physics  of  the  experiment,  instead  of  by  means  of  formulae, 
which  the  average  pupil  will  use  without  verification. 

(3)  Economy  of  time  and  of  effort  is  sought  for  both  teacher 
and  pupil  in  making  the  "  exercise  "  the  unit  of  work.     With 
very  few  exceptions,  each  exercise  can  be  performed  in  a  single 
laboratory  period  of  forty-five  minutes.     Short  experiments  on 
the  same  or  related  topics  are  grouped  into  an  exercise  for 
one  laboratory  period.      This  will  overcome  the  tendency  of 
pupils  to  dawdle  over  short  and  simple  experiments. 


6  PREFACE 

The  order  of  the  chapters  and,  to  some  extent,  of  the  indi- 
vidual exercises,  may  be  varied  to  suit  the  text  or  the  prefer- 
ence of  the  teacher;  but  generally  a  definite  sequence  of 
exercises  within  the  chapters  is  necessary  for  the  logical 
development  of  the  subject,  and,  in  writing  the  manual,  it 
has  been  assumed  that  this  sequence  will  be  observed. 

The  books  referred  to  throughout  the  course  are  the  follow- 
ing: ".A  Brief  Course  in  Physics,77  by  George  A.  Hoadley 
(American  Book  Company)  ;  "  High  School  Physics,77  by  H.  S. 
Carhart  and  H.  N.  Chute  (Allyn  and  Bacon) ;  "  Physics,77  by 
Frederick  Slate  (Macmillan) ;  "  Elements  of  Physics,77  by  Fer- 
nando Sanford  (Holt) ;  "  Heat,  Light  and  Sound,77  by  D.  E. 
Jones  (Macmillan);  "Heat,77  by  H.  ,G.  Madan  (Rivingtons, 
London). 

The  references  are  quite  narrowly  limited  to  the  subject- 
matter  of  the  experiments ;  the  aim  being  to  indicate  the  read- 
ing that  may  profitably  precede  and  accompany  the  laboratory 
work,  without  entering  upon  the  equally  wide  range  of  topics 
which  fall  within  the  scope  of  the  recitation. 

.  The  author  takes  pleasure  in  acknowledging  his  great  in- 
debtedness to  Mr.  George  L.  Leslie,  Head  of  the  Science 
Department  of  the  Los  Angeles  High  School,  under  whose 
helpful  direction  he  obtained  his  first  experience  in  teaching 
physics,  and  whose  laboratory  course  he  has,  during  the  years 
since  1899,  developed  into  the  present  work.  The  author 
regrets  to  state  that  circumstances  rendered  impossible  the 
contemplated  cooperation  of  Mr.  Leslie  in  the  preparation  of 
the  manual  for  publication. 

A  further  acknowledgment  is  due  to  Mr.  A.  G.  Van  Gorder, 
a  former  colleague  at  Los  Angeles,  to  whom  the  author  is 
indebted  for  a  number  of  valuable  suggestions. 

S.   E.   OOLEMAN. 
OAKLAND,  CALIFORNIA. 


CONTENTS 

I.     DIRECTIONS  FOR  LABORATORY  WORK  AND  NOTEBOOK 
KXERCISE  II.     DENSITY  AND  MOLECULAR  PHENOMENA 


*1  .    Density  of  Solids       .         .         .        .         .         .         . 

18 

*2.    Density  of  Liquids     .         .         .        .         .               '  . 

.       20 

*3i.  Cohesion  and  Adhesion     .         .        .         . 

.       21 

32.  Cohesion  and  Adhesion  (alternative)         . 

.       23 

*4.    Surface  Tension  and  Capillarity        

.       25 

III.     MECHANICS  OF  FLUIDS 

*5.    Liquid  Pressure         

,       29 

*6i.  Buoyancy  of  Liquids          .        ... 

,       31 

62.  The  Buoyant  Force  of  Water  (alternative) 

.       34 

*7.    Specific  Gravity  of  Solids  .*       . 

.       35 

*8.    Specific  Gravity  of  Liquids        ...... 

.       37 

9.    Specific  Gravity  of  Liquids        

.       38 

*10.    Gas  Pressure      ....... 

40 

11.    Specific  Gravity  of  a  Liquid      ...... 

.       42 

*12.    The  Siphon  and  the  Suction  Pump    

.       44 

13.    The  Law  of  Boyle     .         .         

.       45 

14.    The  Density  of  Air    

.       48 

IV.     MECHANICS  OF  SOLIDS 

*1  5.    Composition  of  Forces       ....... 

.       51 

16.    Equilibrium  of  Parallel  Forces  ...... 

.       54 

*17.    Moments  of  Force     

.       57 

*18.    Center  of  Gravity  and  Equilibrium  ..... 

.       60 

19.    Equilibrium  and  Stability          ...... 

.       63 

20.    Comparison  of  Masses  by  Inertia      .         .         . 

.       65 

21.    Falling  Bodies  :  Whiting's  Method   .         . 

.       68 

*22.    The  Simple  Pendulum        

.       70 

23.    The  Wheel  and  Axle          

.       73 

*24.    The  Pulley         

.       75 

25.    The  Inclined  Plane    

.       79 

V.     HEAT 

*26.   Expansion  by  Heat   

.       82 

*27.    Conduction  of  Heat  ........ 

.       83 

*28.    Convection  of  Heat   

.       85 

29.    Radiant  Energy         

.       87 

30.    Coefficient  of  Linear  Expansion         

.       90 

31.    Coefficient  of  Expansion  of  Liquids 

93 

32.    Coefficient  of  Expansion  of  Air         ..... 

.       97 

*33.    Melting  and  Freezing.     Solution       

.       99 

*34.    Evaporation.     Vapor  Pressure.     Dew-point 

102 

*35.    Boiling  of  Water        

.     105 

36.    Boiling  Point  of  a  Liquid  

.     107 

CONTENTS 


EXERCISE 

*37.   Specific  Heat     

PAGE 

111 

*38.   Latent  Heat  of  Fusion  of  Ice    

.     113 

39.   Latent  Heat  of  Vaporization  of  Water      .... 

116 

VI.     SOUND 

*40.    The  Transmission  of  Sound       

.     118 

41.    The  Velocity  of  Sound  in  Air   ...... 

120 

42.    The  Reflection  of  Sound    

.     123 

43.   Sympathetic  and  Forced  Vibrations  .... 

125 

*44.    Wave  Length  by  Resonance      

.     128 

45.    Interference  and  Beats      

.     132 

46.   Vibrating  Strings.     Effect  of  Length 

134 

VII.     LIGHT 

*47.    Some  Results  of  Rectilinear  Propagation  .... 

.     136 

*48.    Photometry        

.     139 

*49i.  Plane  Mirrors    

145 

492.  Plane  Mirrors  (alternative)       .        ... 

.     148 

50.    Multiple  Images         ........ 

.     150 

51.    The  Concave  Mirror  

.     153 

*52.    Phenomena  due  to  Refraction   

.     157 

*53.    Index  of  Refraction  ........ 

.     162 

54.    Total  Reflection         

.     166 

*55.    The  Convex  Lens       

.     169 

56.   The  Focal  Length  of  a  Lens      

.     172 

*57.    The  Simple  Microscope     ....... 

.     176 

*58.    Color.        

177 

59.    Spectra      .......... 

180 

*60.   The  Astronomical  Telescope     

.     185 

61.   The  Galilean  Telescope      

.     187 

62,    The  Compound  Microscope        ...... 

.     188 

VIII.     MAGNETISM  AND  ELECTRICITY 

*63.   Magnets  and  Magnetic  Action  

.     191 

*64.   Magnetic  Induction.     Theory  of  Magnetism     . 

.     192 

*65.   The  Magnetic  Field   

.     195 

*66.    The  Single-fluid  Cell          

.     196 

*67.    The  Magnetic  Effects  of  a  Current    

.     198 

68.    The  Tangent  Galvanometer       ...... 

.     204 

69.    The  Laws  of  Resistance     

.     207 

*70.    Measurement  of  Resistance        

.     210 

71.    The  Resistance  of  a  Cell    

.     212 

*72i.  The  Electromotive  Force  of  Cells      

.     214 

722.  The  Electromotive  Force  of  Cells  (alternative) 

.     216 

73.    Arrangement  of  Cells        ....... 

.     217 

*74.    Induced  Currents       

.     219 

*75    The  Motor                                            

222 

*76.    The  Electric  Bell  and  the  Telegraph          .... 

.     224 

77    The  Telephone                                              .... 

226 

APPENDIX  . 

229 

PHYSICAL  LABORATORY  MANUAL 


I.     DIRECTIONS    FOR    LABORATORY    WORK 
AND   NOTEBOOK 

GENERAL   DIRECTIONS 

1.  On   laboratory  days   the  work   required  of  the  student 
outside  the  laboratory  is  :  (1)  the  completion  of  the  record  for 
the  exercise  last  performed ;   (2)  a  careful  reading  of  the  refer- 
ence in  the  text-book  and  of  the  laboratory  directions  for  the 
exercise  for  the  next  day. 

2.  Check  the  list  of  apparatus  before  beginning  any  experi- 
ment, and  report   any  deficiency  to  the   instructor  at   once. 
Never  borrow  apparatus  from  other  places. 

3.  Remember  that  habits  of  neatness  and  order  are  important 
as  well  as  knowledge.     Students  should  feel  a  personal  respon- 
sibility for  the  condition  of  the  apparatus  and  table  where  they 
are  at  work,  and  especially  for  the  condition  in  which  they  are 
left  at  the  close  of  the  exercise.     Learn  to  cooperate  with  the 
instructor  in  keeping  the  laboratory  in  order.     Students  are 
responsible  for   all  damage  to  apparatus,   and  should  report 
such  damage  to  the  instructor  immediately. 

4.  Ordinarily  no  time  is  to  be  taken  in  the  laboratory  for 
writing  discussions  of  experiments  or  for  making  computations, 
except  in  so  far  as  the  results  of  these  computations  are  required 
for  immediate  use ;  but  brief  intervals  that  can  not  be  utilized 
for  experimental  work  should  be  so  spent. 

9 


10        LABORATORY  WORK  AND  NOTEBOOK 

5.  All  questions  asked  in  connection  with  the  experimental 
work  should  receive  the  immediate  and  careful  attention  of  the 
student,  as  an  understanding  of   them  is  necessary  for   the 
intelligent  performance  of  the  experiment.     Ask  the  instructor 
for  help  on  these  questions  when  it  is  necessary. 

THE   RECORD 

6.  At  the  beginning  of  the  record  of  each  exercise  write  its 
number  and  title.     Do  not  copy  the  list  of  apparatus.     Copy 
and  number  all  subheadings.     All  parts  of  the  record  are  to  be 
paragraphed,  numbered,  and  lettered  to  correspond  with  the 
directions. 

7.  All  observations,  computations,  and  results  should  stand 
out  boldly  on  the  page,  separated  from  all  descriptive  and 
explanatory  matter.     In  general,  they  should  be  tabulated,  or 
each  item  should  be  entered  on  a  separate  line. 

8.  Do  not  record  results  in  the  manual.     Measurements 
should  be  immediately  recorded  (with  pencil)  in  proper  form 
in  the  notebook.     A  record  taken  in   any  other  way,  to  be 
copied  into  the  notebook  later,  is  not  recommended. 

9.  Aim  at  brevity  of  statement  without  omitting  essentials. 
Use  only  decimal  fractions  in  the  metric  system,  and  express 
a   quantity  in   terms  of   one  unit   only;    for  example,   write 
15.25  g.  instead  of  15  g    2  dg.  5  eg.     Express  all  lengths  in 
centimeters. 

10.  Your  record  should  be  complete  in  itself ;  that  is,  it  should 
not  require  reference  to  the  directions  <to  make  it  fully  intelli- 
gible. The  "  Form  of  Record'7  is  intended  to  serve  as  a  model 
for  the  tabulation  of  numerical  data  only,  and  is  never  to  be 
regarded  as  indicating  the  complete  record  of  tho  exercise. 


COMPUTATIONS  11 

11.  A  figure  drawn  for  the  .purpose  of  explanation  should,  in 
general,  represent  a  plane  section  through  the  apparatus,  rather 
than  a  view  in  perspective. 

12.  In  the  matter  of  neatness,  as  well  as  of  accuracy,  the 
record  should  be  the  best  that  the  student  is  capable  of.     An 
untidy  page  should  always  be  copied. 

COMPUTATIONS 

13.  All  computations  are  to  be  indicated.     In  the  first  two 
exercises  the  computations  are  indicated  in  the  form  of  record 
as  a  reminder  of  this  direction.     You  are  expected  to  remember 
it   thereafter   without   this   hint.      Where    computations    are 
entered   in   tabular   form    in    columns,   the    headings   of    the 
columns  are  a  sufficient  indication  of  the  operations;   hence 
only  the  results  are  to  be  entered  in  the  columns.     Remember 
that  a  ratio   is    an  indicated   division;    always   perform   the 
division  and  express  the  value  of  the  ratio  decimally. 

14.  All  concrete  numbers  should  be  followed  by  the  name  of 
the  unit  or  the  units  in  which  they  are  measured.     This  applies 
to  all  the  numbers  obtained  in  a  series  of  computations  as  well 
as  to  the  final  result.     A  strict  observance  of  this  direction  will 
be  very  helpful.     Completeness  of  expression  compels  definite- 
ness  of  thought ;  the  want  of  it  is  at  the  root  of  the  greater  por- 
tion of  the  difficulties  that  the  average  student  encounters. 

15.  It  is  important  to  test  the  accuracy  of  numerical  results 
whenever  possible.     This  is  done  by  finding  the  per  cent  of 
error  of  the  result.     What  is  meant  by  this  will  be  seen  from 
the  following  example  :  — 

Suppose  a  length  of  10.05  cm.  is  measured  and  recorded  as 
10  cm.  The  error  is  .05  cm. ;  and  is  .05  -=- 10.05,  or  .005  of  the 
quantity  measured,  or  .5  %  of  it.  If  the  same  error  (.05  cm.) 
were  made  in  measuring  a  distance  of  2.5  cm.,  the  per  cent  of 


12  LABORATORY   WORK   AND   NOTEBOOK 

error  would  be  .05  -4-  2.5  x  100  %,  or  2.5  %.  It  will  be  seen 
that  the  importance  of  a  given  error  is  greater  as  the  quantity 
measured  is  smaller,  and  that  the  degree  of  accuracy  of  a 
measurement  is  expressed  not  by  the  error  but  by  the  per  cent 
of  error.  Pupils  frequently  write  the  per  cent  of  error  100 
times  too  small. 

16.  If  the  per  cent  of  error  of  any  result  is  too  large,  the 
pupil   is   expected   to   repeat   the   experiment  without  delay. 
Only  reasonably   accurate   results   should   be   handed   in   for 
inspection. 

17.  Computations    should   be  carried   to  the  first  doubtful 
figure,  and  all  decimal  places  beyond  this  should  be  dropped, 
as   they  are  wholly  unreliable.      For   example,  suppose   that 
the  weight  of   1   ccm.  of  water,  computed  from  the  measure- 
ments obtained  in  Exercise  2,  is  found  to  be  .99372  g. ;  and 
that  there  is  a  possibility  of  an  error  of  .5  %,  or  about  .005  g. 
In  this  case  the  third  figure  (counting  from  the  left)  is  doubt- 
ful, and  the  result  should  be  written  .994  g. 

18.  Decimal   points   are   frequently  misplaced.     The  most 
casual  inspection  of  the  numbers  is  often  sufficient  to  detect 
such  an  error.     Thus,  if  38.2  be  divided  by  .094,  it  will  be  evi- 
dent at  a  glance  that  the  quotient  is  slightly  greater  than  10 
times  the  dividend  (since  the  divisor  is  slightly  less  than  .1) ; 
and  that,  if  the  quotient  is  written  40.64  or  4064.,  the  decimal 
point  is  misplaced. 

Gross  errors  in  computation  can  often  be  detected  by  the 
absurdity  of  the  results  obtained.  Thus,  if  one  should  obtain 
.412  g.  as  the  weight  of  1  ccm.  of  stone  in  Exercise  1,  he 
should  see  at  once  that  a  serious  blunder  had  been  made 
either  in  the  experimental  work  or  the  computations ;  since 
this  is  less  than  half  the  weight  of  a  cubic  centimeter  of  water 
(1  g.),  and,  for  equal  bulk,  stone  is  much  heavier  than  water. 


COMPUTATIONS  13 

19.    Work   must   never   be   entered   in    such   form    as    the 
following :  — 

5  x  8  x  10  =  400  -  500  =  .8. 

This  equation  expresses  the  absurdity  that  5  x  8  x  10  =  .8. 
Two  equations  are  necessary  to  express  the  relations  intended; 
namely :  — 

5  x  8  x  10  =  400, 

400 -r- 500   =.8. 


20.  Finally.  —  All  the  above  directions  are  important.  Al- 
though some  of  them  can  best  be  understood  in  connection  with 
the  actual  work  of  the  laboratory,  they  are  included  here  for 
convenience  of  reference.  Study  them  carefully  at  the  begin- 
ning, and  look  them  over  frequently  during  the  first  few  weeks, 
until  you  are  sure  you  understand  them  and  know  that  you  are 
following  them. 


II.    DENSITY   AND   MOLECULAR   PHE- 
NOMENA 

MEASUREMENTS 

21.  Measurement  of  Length.  —  The  customary  unit  of  length 
in  scientific  work  is  the  centimeter ;  and  all  lengths  are  to  be 
recorded  in  this  unit  unless  otherwise  specified.     Millimeters 

are  recorded  as  tenths  of  a  centi- 
meter. Fractions  of  a  millimeter 
are  to  be  estimated  in  tenths  and 
recorded  as  hundredths  of  a  centi- 
meter. Thus  the  length  of  the 
block  shown  in  the  figure  is  2.35  cm. 
The  figure  also  shows  the  correct 
position  of  a  meter  rod  in  measur- 
ing. If  the  rod  were  turned  flat, 
the  scale  would  be  at  a  distance  and  the  measurement  would  be 
less  accurate.  If  the  rod  is  worn  at  the  end,  it  is  better  to  begin 
at  some  other  even  centimeter  or,  better  still,  even  decimeter. 

22.  Order  of  trying  Weights  in  Weighing. — Without  a  sys- 
tematic method  of  procedure  in  the  use  of  the  weights  much 
time  is  wasted  in  the  process  of  weighing.     Students  often 
think  that  they  need  more  small  weights  than  are  provided, 
and  resort  to  the  forbidden  practice  of  borrowing.     A  full  set 
of  weights,  such  as  you  will  always  find  provided,  includes  all 
that  are  necessary  to  balance  any  weight  from  one  equal  to  the 
sum  of  all  the  weights  of  the  set  down  to  one  as  light  as 
the  smallest  of  the  set.     But  a  single  set  is  adequate  only 
when  the  weights  are  tried  in  proper  order. 

14 


MEASUREMENTS  15 

Begin  by  trying  the  weight  which  you  estimate  to  be  nearest 
to  the  weight  to  be  balanced.  If  it  seems  to  be  nearly  suffi- 
cient, add  the  next  smaller;  but  if  it  is  evidently  much  too 
light,  replace  it  and  try  the  next  larger.  Proceed  thus  back- 
ward, trying  the  larger  weights  till  you  are  assured  that  the 
next  larger,  used  alone,  is  too  large.  The  secret  of  rapid 
iveighing  is  to  try  out  the  larger  weights  first.  If  the  process  is 
begun  with  too  small  a  weight,  this  is  commonly  not  discovered 
till  all  the  smaller  weights  have  been  added,  when  the  whole 
process  must  again  be  repeated ;  whereas  it  would  have  been 
discovered  by  trying  the  single  larger  weight. 

Having  thus  determined  the  largest  weight  to  be  used,  add 
the  smaller  weights  in  succession.  If  any  weight  proves  to  be 
too  great  an  addition,  remove  it  (not  a  larger  one)  and  try  the 
next  smaller.  Continue  thus  till  you  come  to  the  smallest 
weight  provided. 

23.  Use  of  the  Platform  Balance.  —  The  platform  balance  is 
provided  with  a  graduated  beam,  which  is  to  be  used  instead 
of  weights  smaller  than  the  maximum  reading  of  the  beam 
(usually  5  g.).  In 
using  a  platform  bal- 
ance it  must  be  ob- 
served that  the  beam 
adjustment  acts  with 
the  weights  or  with  the 
article  weighed,  depend- 
ing upon  whether  the 
article  to  be  weighed  FlG  2 

is  placed  on  the  one 

platform  or  the  other.  Make  the  beam  adjustment  act  with  the 
weights,  and  add  the  beam  reading  to  the  sum  of  the  weights 
used. 

Proceed  as  follows :  Place  the  article  to  be  weighed  on  the 
proper  platform.  Slide  the  weight  on  the  beam  to  the  zero 


DENSITY  AND  MOLECULAR  PHENOMENA 


end.  The  beam  adjustment  is  to  be  held  in  reserve  till  the 
weighing  has  been  brought  within  the  limit  of  the  beam 
reading.  When  this  has  been  accomplished,  make  the  final 
adjustment  with  the  beam.  You  can  shorten  this  process  by 
steadying  the  platforms  with  the  hands.  Do  not  waste  time 
waiting  for  the  oscillations  to  cease  entirely.  Some  platform 
balances  are  provided  with  a  vertical  pointer  between  the 
platforms.  If  the  oscillations  are  small  and  this  pointer 
moves  to  approximately  equal  distances  on  both  sides  of  the  zero 
point  of  the  graduated  arc  behind  it,  the  weighing  is  sufficiently 
exact.  The  weight  of  the  object  is  the  sum  of  the  weights 
used  plus  the  beam  reading.  This  balance  is  hardly  sensitive 
to  less  than  .1  g.,  and  readings  to  this  fraction  are  sufficient. 

24.  Use  of  the  Specific  Gravity  or  Beam  Balance.  —  The  beam 
of  this  balance  is  not  fastened  to  the  upright,  and  is  easily 
thrown  out  of  place  in  putting  heavy  objects  on  or  removing 

them  from  the  pans.  To  avoid 
this,  always  support  the  pan  on 
the  heavier  side  by  placing  a  hand 
under  it.  When  balance  is  nearly 
secured,  time  can  be  saved  by* 
steadying  the  pans  with  the  hands. 
Watch  the  vertical  pointer  carried 
by  the  beam,  and  adjust  the 
weights  according  to  its  indication. 
The  weighing  is  sufficiently  exact 
when  the  distances  to  which  the 
pointer  swings  on  each  side  of  the 

zero  differ  by  less  than  one  division  of  the  scale  behind  it. 

With  this  balance,  weight  should  be  determined  to  the  nearest 

centigram. 

25.  Precautions    to    be    observed    in    Weighing.  —  Weights, 
especially  heavy  ones,   should  be  placed,  near  the  center  of 
the  platform  or  pan. 


Fia.  3. 


MEASUREMENTS  17 

It  is  better  to  handle  weights  with  forceps  than  with  the 
fingers.  If  forceps  are  provided,  use  them.  You  will  find 
them  more  convenient  than  the  fingers,  especially  in  handling 
the  fractional  weights. 

Always  return  weights  to  the  proper  places  in  the  block  as 
soon  as  you  have  finished  weighing.  The  most  convenient 
method  of  counting  weights  is  to  add  them  up  as  you  return 
them  to  the  block,  beginning  with  the  largest  and  taking  them 
in  the  order  of  their  size.  The  weights  should  never  be  put 
down  anywhere  except  on  the  balance  and  in  their  proper 
places  in  the  block.  Remember  that  if  one  of  the  weights  is 
lost  the  whole  set  is  practically  useless. 

When  fractional  weights  are  provided,  they  should  consist 
of  the  following:  .5,  .2,  .1,  .1,  .05,  .02,  .02,  and  .01  g.  If  any 
are  missing,  report  the  fact  to  the  instructor. 

Before  using  a  balance  observe  whether  the  beam  swings 
freely  and  comes  to  rest  in  a  horizontal  position.  If  it  does 
not,  a  bearing  is  probably  out  of  adjustment. 

In  weighing  liquids,  see  that  the  outside  of  the  vessel  is  dry 
before  placing  it  on  the  balance.  If  any  liquid  is  spilled,  wipe 
it  up  at  once. 

26.  The  Estimation  of  Tenths.  —  In  all  laboratory  measure- 
ments it  should  be  remembered  that  the  student  is  expected  to 
be  as  accurate  as  it  is  possible  for  him  to  be  with  the  apparatus 
provided.  In  determining  the  position  of  points  on  scales  of 
various  sorts,  —  as  the  position  of  the  end  of  a  line  on  a  meter 
rod,  the  point  on  its  scale  to  which  a  hydrometer  sinks  in  a 
liquid,  the  height  of  the  mercury  on  a  thermometer  scale,  the 
position  of  the  pointer  on  the  scale  of  a  spring  balance,  etc.,— 
the  scale  should  be  read  to  tenths  of  its  smallest  division.  Even 
if  the  student  has  not  the  skill  to  do  this  accurately,  the  esti- 
mate will  be  considerably  more  accurate  than  the  nearest  whole 
division ;  and  it  is  the  universal  rule  that  no  error  should  be 
unnecessarily  introduced  into  the  work. 
COLEMAN'S  PHY.  LAB.  MAN.  —  2 


18 


DENSITY   AND   MOLECULAR   PHENOMENA 


EXERCISE   1.     DENSITY   OF   SOLIDS 

I.  To  find  the  weight  of  one  cubic  centimeter  of  a 
rectangular  block  of  wood. 

Apparatus. — A  rectangular  block  of  wood;  platform  balance; 
weights ;  forceps  for  handling  weights ;  metric  rule. 

[A  30  cm.  metric  rule  is  more  convenient  than  a  meter  stick  for  this 
and  several  other  experiments.] 

Definition. —  The  density  of  a  substance  is  the  mass  of  a  unit 
volume  of  -the  substance.  In  the  metric  system  it  is  usually 
expressed  as  the  number  of  grams  in  a  cubic  centimeter  of  it 
(g.  per  ccm.);  in  the  English  system,  as  the  number  of  pounds 
in  a  cubic  foot  of  it  (Ib.  per  cu.  ft.). 

Measure  with  the  metric  rule  the  dimensions  of  the  block. 
(Observe  the  directions  of  Arts.  21  and  26.)  For  each  dimen- 
sion take  four  measurements  —  one  near  each  of  the  four 
parallel  edges  —  and  take  their  average. 

Weigh  the  block.  (Observe  the  directions  of  Arts.  22,  23,  and 
25.)  Does  it  matter  on  which  platform  you  place  the  block? 
Can  the  graduated  beam  at  the  side  be  used  in  the  weighing 
when  the  block  is  on  either  platform  ?  Always  return  the 
weights  to  their  proper  places  in  the  block  as  soon  as  you  have 
finished  weighing. 

Compute  the  volume  of  the  block  and  the  weight  of  1  ccm. 
of  it  (its  density). 

FORM  OF  KECORD 

DIMENSIONS  OF  THE  BLOCK 


LENGTH 

WIDTH 

THICKNESS 

cm. 

cm. 

cm. 

Av.  cm. 

cm. 

cm. 

MEASUREMENTS  19 

Weight  of  the  block  =  g. 

Volume  of  the  block  =  (    )x(    )x(    )  =  ccm. 

Density  of  the  block  =  (    )  ~  (    )  =  g.  per  ccm. 

IIo  To  find  the  density  of  a  stone  or  other  irregular 
body  that  sinks  in  water. 

Apparatus.  —  Platform  balance  and  weights ;  overflow  can ; 
tumbler ;  vessel  of  water ;  cubic  centimeter  graduate ;  stone 
or  other  solid,  with  string  tied  to  it ;  mop  cloth. 

[The  copper  boiler  used  in  several  of  the  heat  experiments  serves  very 
well  for  an  overflow  can  if  a  short  piece  of  rubber  tubing  is  attached  to 
the  spout  (Fig.  4).] 

Weigh  the  solid.     (Keturn  the  weights  to  the  block.) 

If  the  spout  of  the  overflow  can  is  not  high  enough  to  allow 
the  graduate  to  be  placed  under  it,  set  the  can  near  the  edge  of  the 
table  with  the  spout  projecting  over  the  edge.  Fill  the  can  till 
the  water  begins  to  run  out  of  the  spout,  and  catch  the  over- 
flow in  the  graduate ;  then  empty  the  latter  and  again  hold  it 
under  the  spout.  Lower  the  solid  into  the  can  by  means  of 
the  string,  and  catch  the  overflow  in  the  graduate.  What  is 
the  volume  of  the  overflow  ?  In  measuring  the  water,  hold  the 
eye  on  a  level  with  its  surface,  and  observe 
where  the  lowest  part  of  the  surface  (seen 
through  the  elevated  ridge  of  water  at  the 
edge)  comes  to  on  the  scale.  Estimate  to 
tenths  of  the  smallest  division  on  the  scale. 

How  does  the  volume  of  the  overflow 
compare  with  the  volume  of  the  solid  ? 

Record  each  item  on  a  separate  line,  and 
compute  the  density  of  the  solid.  Indicate 
the  computation. 

If   the  density  of   the  solid  is  given  in 
Table  I  of  the  Appendix,  compute  the  per  cent  of  error  of  your 
result. 


20  DENSITY    AND    MOLECULAR    PHENOMENA 

EXERCISE   2.     DENSITY  OF  LIQUIDS 

I.    To  find  the  density  of  water. 

Apparatus.  —  Beam  balance  and  set  of  weights  to  1  eg.  (or 
platform  balance  and  weights  to  5  g.)  ;  forceps ;  beaker ;  grad- 
uate with  cubic  centimeter  scale ;  mop  cloth. 

a.  Weigh  the  beaker  empty.     (See  Arts.  24  and  25.) 

b.  Fill  the  beaker  nearly  full  of  water  and  weigh  again. 
Always  record  the  quantity  measured;  in  this  case  it  is  the 
weight  of  the  beaker  and  water.     The  weight  of  the  water  is 
found  indirectly  by  subtraction.     The  record  should  be  as  in- 
dicated below.    By  recording  the  actual  observations  it  is  easy 
to  determine  whether  an  error  in  the  final  result  is  due  to  an 
error  in  the  experimental  work  or  in  the  computations. 

c.  Measure  in  the  graduate  the  volume  of  the  water  weighed. 

d.  Compute  the  density  of  the  water  in  grams  per  cubic  cen- 
timeter.    The  density  of  water  at  4°  C.  is  1  g.  per  cubic  centime- 
ter.     At  the  temperature  of  the  laboratory  (about  20°)  it  is 
.998  g.  per  cubic  centimeter.      Assuming  the  latter  to  be  the 
true  value,  find  the  error  (the  difference  between  the  true  value 
and  your  result)  and  the  per  cent  of  error  (the  per  cent  that 
this  difference  is  of  .998).    The  per  cent  of  error  should  be  less 
than  1%. 

Suggest  possible  reasons  why  your  result  is  not  perfectly 
correct. 

FORM  OF  EECORD 

a.  Weight  of  beaker  = g. 

b.  Weight  of  beaker  and  water  = g. 

Weight  of  water  =()  —  ()  = g. 

c.  Volume  of  the  water  = ccm. 

d.  Computed  density  of  water  =  ()-•-()  =  —  —  g.  per  ccm. 
True  value  of  density  (at  20°  C.)  =  .998  g.  per  ccm. 
Error  =()-()  -  g. 

Per  cent  of  error  =  ()-«-()  x  100%   = %. 


COHESION   AND   ADHESION  21 

II.  To  -find  the  density  of  a  saturated  solution  of 
table  salt. 

Apparatus.  —  Beam  balance  and  set  of  weights  to  1  eg.  (or 
platform  balance  and  weights  to  5  g.)  ;  bottle  with  glass  stop- 
per; bottle  of  water;  bottle  of  a  saturated  solution  of  table 
salt  (or  of  other  liquid) ;  jar  of  water ;  beaker ;  mop  cloth. 

a.  Weigh  the  bottle  empty  with  the  stopper. 

b.  Fill  it  with  water  from  the  supply  bottle  and  insert  the 
stopper,  being  careful  to  avoid  a  bubble  of  air  beneath  the  stop- 
per.    Wipe  the  bottle  dry  on  the  outside  and  weigh  it. 

c.  Compute  the  weight  of  water  in  the  bottle.     What  is  the 
capacity  of  the  bottle  in  cubic  centimeters,  assuming  the  density 
of  water  to  be  1  g.  per  cubic  centimeter  ? 

d.  Empty  the  bottle  (into  the  supply  bottle  of  water) ;  fill 
with  the  salt  solution  (or  other  liquid  provided),  exercising  the 
same  precautions  as  before,  and  weigh.     Return  the  solution 
to  the  supply  bottle  and  rinse  the  bottle  in  the  jar  of  water. 
Use  the  mop  cloth,  and  leave  the  table  and  scale  pans  dry,  and 
everything  in  order. 

e.  Compute  the  weight  of  the  salt  solution  used,  and  from 
this  and  its  volume  (the  capacity  of  the  bottle)  compute  its 
density.     Record  each  item  on  a  separate  line. 


EXERCISE   ,%     COHESION   AND  ADHESION 

References.  —  Hoadley,  28 ;  Carhart  and  Chute,  17-18. 

To  study  the  effect  of  distance  upon  molecular  at- 
traction, and  to  determine  the  relative  strength  of 
cohesion  and  adhesion  for  a  number  of  substances. 

Apparatus.  —  Two  pieces  of  plate  glass ;  two  pieces  of  com- 
mon window  glass  ;  piece  of  glass  coated  with  paraffine ;  vessel 
of  water;  small  bottle  containing  a  little  mercury  and  some 
very  small  bits  of  glass ;  dropping  tube. 


22  DENSITY   AND    MOLECULAR   PHENOMENA 

a.  See  that  the  surfaces  of  the  pieces  of  plate  glass  are  clean ; 
then  press  them  firmly  together.     Now  lift  the  upper  piece, 
and  see  whether  the  lower  one  will  stick  to  it  and  be  lifted  with 
it.     Keep  one  hand  in  position  to  catch  the  lower  glass  if  it 
falls.    Pull  the  pieces  apart  after  pressing  them  firmly  together, 
and  note  the  force  with  which  they  stick  together,  or  cohere. 
Describe  briefly  the  observed  effects. 

b.  Experiment  in  the  same  way  with  the  pieces  of  window 
glass.     The  surfaces  of  the  plate  glass  are  quite  accurately 
plane ;  those  of  the  window  glass  are  much  less  so,  in  fact,  are 
distinctly  wavy.     The  difference  in  the  results  of  the  experi- 
ments is  due  to  this  difference  in  the  surfaces.      Hence  the  ex- 
periments indicate  a  necessary  condition  for  the  existence  of 
cohesion.     State  this  condition,  and  show  how  it  follows  from 
the  experiments. 

c.  Again  take  the  pieces  of  plate  glass,  dip  them  in  water  to 
moisten  their  surfaces,  and  press  them  together  as  before.    Now 
pull  them  apart.     Compare  the  forces  involved  with  the  forces 
when  the  plates  were  dry.    The  water  forms  a  connecting  link, 
adhering  to  the  glass  surface  on  each  side. 

Do  you  conclude  from  this  that  adhesion  between  water  and 
glass  molecules  is  greater  than  cohesion  between  two  glass 
molecules  when  the  molecules  are  equally  near  ? 

If  your  conclusion  is  different,  state  it  and  give  reason.  (On 
this  point  consider  whether  it  is  easier  to  wipe  water  from 
a  glass  surface  or  to  wipe  glass  molecules  from  a  glass  sur- 
face.) 

d.  From  the  fact  that  water  wets  glass,  what  do  you  infer 
concerning  the  relative  strength  of  the  molecular  force  between 
two  water  molecules  and  that  between  a  water  and  a  glass 
molecule,  the  molecules  being  the  same  distance  apart  in  both 
cases  ? 

e.  Caution.  —  In  handling  mercury,  be  careful  not  to  get  any 
of  it  on  jewelry.     It  unites  with  gold  and  silver,  forming  an 
amalgam  that  discolors  the  surface. 


COHESION   AND   ADHESION  23 

By  means  of  the  dropping  tube  transfer  a  small  drop  of 
mercury  from  the  bottle  to  a  piece  of  glass.  Flatten  the  drop 
with  the  finger,  then  observe  the  effect  of  removing  the  finger. 

Why  does  the  drop  not  flatten  out  into  a  thin  layer  as  the 
result  of  its  own  weight  ? 

Why  does  it  not  spread  over  the  surface  as  a  drop  of  water 
would  ? 

With  the  point  of  your  pencil  detach  a  very  small  bit  from 
the  drop.  Is  this  smaller  portion  more  or  less  nearly  spheri- 
cal than  the  larger  portion  ?  Why  ?  Pour  the  drop  of  mer- 
cury into  your  hand  and  return  it  to  the  bottle. 

/.  You  may  have  concluded  from  the  preceding  experiment 
that  there  is  no  adhesion  between  glass  and  mercury.  To  test 
this  point,  turn  the  bottle  of  mercury  about  and  note  the 
behavior  of  the  bits  of  glass  toward  the  mercury.  Describe 
their  behavior  and  state  conclusion.  Your  conclusion  in  regard 
to  cohesion  in  mercury  and  adhesion  between  mercury  and 
glass  should  be  consistent  with  the  preceding  experiment  as 
well  as  with  this. 

g.  Study  and  describe  the  behavior  of  a  drop  of  water  on 
the  paraffined  surface  of  the  piece  of  glass. 

Do  you  find  cohesion  within  water  stronger  or  weaker  than 
adhesion  between  water  and  paraffine  ? 

Do  you  infer  that  adhesion  between  water  and  paraffine  is 
greater  or  less  than  between  water  and  glass  ? 


EXEKCISE   32.     COHESION   AND  ADHESION 

(ALTERNATIVE  WITH  EXERCISE  3j) 
References.  —  Hoadley,  28 ;  Carhart  and  Chute,  17-18. 

To  measure  the  force  of  cohesion  or  adhesion  that 
must  be  overcome  in  pulling  a  glass  disk  from  water 
and  from  mercury,  and  a  copper  disk  from*  mercurj/. 


24  DENSITY   AND   MOLECULAR   PHENOMENA 

Apparatus.  —  Specific  gravity  balance ;  weights  to  1  eg. ;  fine 
shot  or  coarse  sand ;  disk  or  square  of  glass  about  5  cm.  in 
diameter,  with  3  or  4  threads  attached  for  suspending  it 
(Fig.  5) ;  disk  or  square  of  copper  of  same  size,  amalgamated 
with  mercury,  with  threads  for  suspension;  tumbler  of  clean 
water ;  tumbler  containing  some  clean  mercury. 

[To  amalgamate  the  copper,  put  it  in  a  dish  containing  a  little  mercury 
and  dilute  sulphuric  acid,  and  rub  the  acid  and  mercury  over  its  surface 
with  a  rag  tied  to  a  stick.  Mercury  will  wet  such  an  amalgamated  sur- 
face if  it  is  kept  clean.  Avoid  touching  it  with  the  fingers.  ] 

a.  Suspend  the  glass  disk  from  the  hook  on  the  under  side 
of  the  higher  scale  pan ;  and  adjust  the  standard  of  the  balance 
or  the  glass  of  water  so  that  the  disk  just  touches  the  water 
when  the  beam  is  horizontal.     The  disk  must  be  exactly  hori- 
zontal when  all  the  cords  are  stretched ;   but,  to 
avoid  air  bubbles  under  it,  touch  one  side  of  it  to 
the  water   first,  then  gradually  lower   the   other. 
Pour  the  shot  (or  sand)  slowly  into  the  scale  pan 
on  the  opposite  side  till  the  disk  is  torn  away  from 
the  water.      Eemove  the  tumbler   of  water,  and 
restore    equilibrium    by   placing   weights    in    the 

scale  pan  from  which  the  disk  is  still  suspended.  These 
weights  measure  the  force  exerted  in  tearing  the  disk  from 
the  water. 

b.  Is  the  under  surface  of  the  disk  dry  or  wet  ?     Was  the 
force  that  was  overcome  adhesion  between  the  disk  and  the 
water  or  cohesion  between  a  thin  layer  of  water  in  contact 
with  the  disk  and  the  water  beneath  ? 

Which  is  the  stronger,  cohesion  in  water  or  adhesion  between 
water  and  glass  ? 

c.  In  the  same  way  measure  the  force  necessary  to  tear  the 
glass  disk  from  the  mercury.     The  glass  must  be  dry  and  the 

'surface  of  the  mercury  very  clean.  Impurities  on  the  surface 
may  be  removed  by  skimming  it  with  a  bit  of  paper  pushed 
into  the  mercury  and  drawn  across  the  surface. 


SURFACE   TENSION   AND   CAPILLARITY  25 

d.  Did  the  mercury  wet  the  disk  ?     Which  joint  was  broken 
when  the  disk  was  torn  away,  the  adhesion  joint  between  glass 
and  mercury,  or  the  cohesion  joint  between  adjacent  layers  of 
mercury  ? 

e.  Measure  in  the  same  way  the  force  necessary  to  tear  the 
amalgamated  copper  disk  from  the  mercury.     Do  not  touch 
the  under  surface  of  the  disk  with  the  lingers ;  the  oil  from 
them  would  atfect  the  adhesion  of  the  mercury  to  the  surface. 
If  the  disk  is  clean,  as  it  should  be,  its  surface  will  be  wet 
with  mercury  when  it  is  torn  away. 

Have  you  measured  cohesion  in  mercury  or  adhesion  between 
mercury  and  the  amalgamated  surface  of  the  copper  ? 

Discussion.  —  a.  Which  is  stronger,  adhesion  between  glass 
and  mercury,  or  cohesion  in  mercury  ?  Show  how  your  answer 
follows  from  the  results  of  this  exercise. 

b.  Arrange  in  their  order  of  magnitude,  cohesion  in  mer- 
cury, cohesion  in  water,  adhesion  between  mercury  and  glass, 
and  adhesion  between  the  amalgamated  copper  and  mercury. 
What  do  you  know  of  the  position  of  adhesion  between  water 
and  glass  in  this  list  ? 

EXERCISE  4.     SURFACE   TENSION   AND 
CAPILLARITY 

References.  —  Hoadley,  119-126;  Carhart  and  Chute,  113- 
122  ;  Slate,  80-82  ;  Sanford,  pp.  102-103. 

I.    To  study  various  phenomena  due  to  surface  tension. 

Apparatus. — Pins;  tumbler  of  clean  water;  lifter  made  of 
wire,  for  placing  pins  on  water  (Fig.  6) ;  small  bottle  of  alco- 
hol ;  glass  rod  ;  small  rubber  band  ;  dish  of  soap  solution  ; 
wire  ring  about  3  in.  in  diameter,  with  extension  of  wire  for 
handle,  and  with  a  thread  tied  across  the  ring  and  carrying  a 
loop  at  its  middle  (Fig.  7) ;  bottle  containing  a  drop  of  oil  sus- 
pended in  a  solution  of  alcohol  and  water  of  its  own  density. 


26 


DENSITY   AND   MOLECULAR   PHENOMENA 


FIG.  6. 


[To  prepare  the  suspended  drop  of  oil,  pour  a  very  little  water  into 
the  bottle ;  add  two  or  three  drops  of  oil ;  then  pour  in  alcohol  till 
the  oil  sinks  part  way.  Tip  the  bottle  and  pour  the  alcohol  in  gently 
to  avoid  breaking  the  oil  up  into  minute  globules.] 

a.  Place  a  pin  on  the  lifter  (Fig.  6)  and  lower  it  slowly  till 

the  pin  floats  on  the  water,  then  remove 
the  lifter  carefully.  If  the  pin  sinks  now 
or  later  during  the  experiment,  take  it  out 
with  the  lifter,  wipe  it  dry,  and  try  again. 
Observe  closely  the  shape  of  the  surface  of 
the  water  about  the  pin.  Describe  it.  Draw 
an  enlarged  figure  of  a  cross  section  of  the 
pin  and  adjacent  water  surface. 

Push  the  pin  till  it  breaks  through  the 
surface.     Why  does  it  sink  now  ?     Why  did  it  float  before  ? 

b.  Float  two  pins  on  the  water,  placing  them  parallel  and 
about  2  cm.  apart.     Dip  the  glass  rod  into  the  alcohol,  and 
carry  a  drop  on  the  end  of  it  to  the   surface  of  the  water 
between   the  pins,  at   the  same  time  watching  the  behavior 
of  the  pins.     (If  you  do  not  succeed  in 

the  experiment  after  a  few  trials,  you 
may  substitute  the  rubber  band  for  the 
pins,  and  drop  the  alcohol  inside  of 
it.)  What  happens  ?  In  trying  to 
understand  what  happens,  think  of  the 
water  surface  as  a  stretched  membrane, 
attached  (by  adhesion)  to  both  sides 
of  the  pin  and  pulling  it  equally  on 
both  sides. 

What  is  the  effect  of  the  alcohol  on 
the  surface  tension  or  stretch  ? 

c.  Dip  the  wire  ring  into  the  soap  solution  and  withdraw  it 
obliquely.     Repeat,  if  necessary,  till  a  film  is  obtained.     Hold 
the  ring  so  that  the  loop  of  thread  will  attach  itself  to  the 
film  ;  then,  with  the  point  of  your  pencil,  break  the  film  inside 


FIG.  7. 


SURFACE    TENSION   AND   CAPILLARITY  27 

the  loop.     Describe  and  explain  the  behavior  of  the  loop  when 
the  film  inside  it  is  broken. 

d.  Describe  the  appearance  of  the  oil  globule  suspended  in 
the  solution  of  alcohol  and  water.  An  inflated  rubber  balloon 
and  a  soap  bubble  take  the  same  shape  for  similar  reasons. 
Describe,  compare,  and  explain.  Shake  the  bottle  very  gently 
so  as  to  distort  the  drop,  and  note  its  behavior.  Be  careful 
not  to  shake  so  hard  as  to  break  up  the  drop.  Describe  and 
explain  its  motion. 

II.  To  study  capillary  action  with  reference  to  the 
effect  of  the  size  of  the  tubes  and  the  relative  strength 
of  cohesion  and  adhesion. 

Apparatus. — Capillary  tubes  of  glass  of  different  sizes; 
capillary  tube  coated  inside  with  paraffine ;  glass  of  water ; 
glass  containing  a  little  mercury ;  blotting  paper. 

[To  coat  a  tube  with  paraffine,  put  a  very  small  piece  of  paraffine  in 
one  end  of  the  tube  ;  hold  it  obliquely  in  a  flame  till  the  paraffine  melts 
and  runs  down  the  tube.] 

a.  Observe  the  shape  of  the  water  and  mercury  surfaces 
near  the  glass.     Draw  figures  of  vertical  sections  through  the 
liquids  to  illustrate.     Account  for  the  difference  in  the  shape 
of  the  two  surfaces. 

b.  Do  not  use  the  same  tubes  in  the  water  and  the  mercury. 
Insert  tubes  of  different  sizes  into  the  water,  and  observe  the 
rise  of  the  water  in  the  tubes.     The  forces  that  produce  this 
motion  are  the  molecular  forces  of  cohesion  and  adhesion  that 
were  studied  in  the  preceding  exercise.     Describe  the  shape  of 
the  water  surface  in  the  tubes.     Remember  that  the  surface  is 
under  a  tension,  and  is  attached  by  adhesion  to  the  glass. 

Explain  how  the  rise  of  the  water  is  brought  about. 

Draw  a  figure  in  cross  section  showing  the  height  of  the 
water  and  the  shape  of  the  surface  in  the  tumbler  and  in  tubes 
of  different  sizes  inserted  in  it. 


28  DENSITY   AND   MOLECULAR   PHENOMENA 

c.  Kepeat  the  preceding  with  tubes  of  different  sizes  in  mer- 
cury, using  only  dry  tubes.     By  pressing  the  tube  against  the 
side  of  the  glass,  an  unobstructed  view  of  the  inside  of  the  tube 
will  be  obtained.     Describe  what  is  seen  and  draw  figures  to 
illustrate. 

d.  The  rise  of  the  water  and  the  fall  of  the  mercury  in  the 
tubes  are  due  to  the  direction  of  curvature  of  their  surfaces. 
Explain  how. 

What  determines  the  direction  of  curvature  of  the  surfaces  ? 

Test  these  matters  further  by  trying  in  water  the  capillary 
tube  coated  on  the  inside  with  paraffine.  Describe  and  account 
for  the  result,  bearing  in  mind  the  behavior  of  the  drop  of 
water  on  the  piece  of  glass  covered  with  paraffine  in  Ex- 
ercise 3!. 

e.  Pour  a  few  drops  of  water  upon  the  mercury,  and  remove 
it  with  the   blotting  paper.     Observe   the   difference   in   the 
behavior  of  the  water  and  the  mercury  toward  the  blotting 
paper.     Describe  and  explain. 


III.     MECHANICS   OF   FLUIDS 
EXERCISE  5.     LIQUID  PRESSURE 

References.  —  Hoadley,  131-142 ;  Carhart  and  Chute,  123- 
133  ;  Slate,  45-47. 

I.  To  observe  the  effect  of  the  depth  of  a  liquid  upon 
the  pressure  exerted  by  it. 

Apparatus  (for  Parts  I  and  II).  —  Battery  jar  filled  with 
water ;  Welsbach  or  student  lamp  chimney ;  small  square  of 
cardboard ;  stick  1  ft.  long ;  two  beakers,  one  low  and  wide, 
the  other  tall  and  slender,  of  about  the  same  weight,  but  very 
unequal  diameters. 

a.  Hold  the  cardboard  over  an  end  of  the  chimney,  and 
lower  this  end  into  the  jar  of  water.     Remove  the  hand  from 
the  cardboard  when  you  find  that  it  will  remain  in  place  with- 
out being  held.     What  keeps  it  in  place  ? 

b.  With  the  -cardboard  covering  the  lower  end  of  the  chim- 
ney, lower  it  till  it  no  longer  tends  to  sink.     Steady  it,  but  do 
not  support  it  with  the  hands.     What  sustains  the  chimney  ? 
It  may  help  you  to  understand  this  if  you  put  the  stick  down 
through  the  chimney  and  push  the  cardboard  from  the  end  of 
it,  at  the  same  time  noting  the  force  necessary  to  do  so. 

c.  Replace  the  cardboard  over  the  end  of  the  chimney,  and 
push  the  latter  slowly  into  the  water,  at  the  same  time  noting 
the  change  in  the  force  you  must  exert  to  keep  it  in  place. 
State  any  relation  that  you  discover  between  the  depth  and  the 
force  (pressure)  exerted  by  the  water.     Do  not  make  your  state- 
ment more  explicit  than  the  crude  nature  of  your  experiment 
will  warrant.     The  exact  relation  between  depth  and  pressure 

29 


30  MECHANICS   OF   FLUIDS 

can  be  obtained  experimentally  only  -  by  careful  measurement 
of  the  depths  and  pressures. 

What  is  the  direction  of  the  pressure  you  have  been  investi- 
gating ? 

II.  To  observe  whetlier  the  total  pressure  is  affected  by 
the  area  of  the  surface  pressed  upon. 

Push  the  two  beakers,  one  in  each  hand,  bottom  down,  into 
the  water  to  the  same  depth ;  and  notice  which  requires  the 
greater  force  to  keep  it  in  place.  Explain. 

III.  To  observe  the  effect  of  the  density  of  the  liquid 
upon  the  pressure  exerted  by  it. 

Apparatus. — Tumbler  containing  mercury ;  tumbler  of  water ; 
two  blocks  of  the  same  size  and  shape  and  small  enough  to  go 
into  the  tumblers  ;  chunk  or  ball  of  iron. 

a.  Lift  the  tumbler  of  mercury ;  handle  it  carefully.     Lift 
the  tumbler  of  water.     Mercury  is  13.6  times  as  dense  as  water. 
Float  one  of  the  blocks  on  the  water  and  the  other  on  the  mer- 
cury.    Account  for  the  difference  in  the  depths  to  which  the 
two  blocks  sink. 

b.  Push  the  blocks  down  till  they  are  submerged  in  the 
liquids.     Compare  the  forces  necessary  to  do  this,  and  account 
for  their  difference. 

c.  Put  the  chunk  or  ball  of  iron  into  the  mercury.     What 
happens  to  it  ?     Explain. 

IV.  To  measure  liquid  pressure  with  a  pressure  gauge, 
or  manometer. 

Apparatus.  —  Hydrometer  jar  of  water  ;  a  water  and  a  mer- 
cury manometer ;  metric  rule. 

a.  A  bent  tube  containing  a  liquid  and  used  to  measure  pres- 
sures exerted  by  liquids  or  gases  is  called  a  pressure  gauge  or 
manometer  (Fig.  8).  Lift  the  water  manometer  out  of  the  jar. 


BUOYANCY   OF  LIQUIDS  31 

Compare  the  level  of  the  water  in  the  two  arms  of  the  bend. 
Slowly  lower  the  manometer  into  the  jar  of  water,  and  observe 
the  behavior  of  the  water  in  the  bend  of  the  tube. 
Describe  and  account  for  its  motion. 

How  is  the  pressure  of  the  water  in  the  jar  trans- 
mitted to  the  water  in  the  manometer  ? 

While  moving  the  manometer  up  and  down,  com- 
pare the  difference  of  level  of  the  water  in  the 
two  arms  of  the  bend  with  the  difference  between 
the  level  of  the  water  in  the  jar  and  in  the  lower 
end  of  the  manometer.  Make  two  or  three  exact 
measurements  of  these  distances  with  the  manom- 
eter at  different  depths.  How  do  they  compare  ? 

b.    Repeat  with  the  mercury  manometer,  except 
the  measurements,  which  need  not  be  taken.     State 
the  points  of  similarity  with  the  preceding.     State     ' 
and  account  for  the  differences. 


EXERCISE    G!.     BUOYANCY  OF  LIQUIDS 

References.  —  Hoadley,  143-144;  Carhart  and  Chute,  134- 
138. 

Apparatus.  —  Specific  gravity  balance  and  weights  to  1  eg., 
or  platform  balance  and  weights  to  5  g. ;  for  the  platform 
balance  a  support  as  shown  in  Fig.  9 ;  overflow  can ;  beaker 
or  tumbler;  jar  of  water;  jar  of  solution  of  table  salt;  stone; 
block  of  wood ;  mop  cloth. 

I.  To  find  the  relation  between  the  loss  of  weight  of 
a  stone  in  water  and  the  weight  of  the  water  that  it 
displaces. 

a.  Weigh  the  stone. 

b.  Weigh  the  beaker,  and  place  it  under  the  spout  of  the 
overflow  can.    Fill  the  can  with  water  till  it  begins  to  overflow. 


32 


MECHANICS   OF   FLUIDS 


Empty  the  beaker,  replace  it  under  the  spout,  lower  the  stone 

into  the  can  by  means 
of  a  thread,  and  catch 
beaker 


in     the 


and 


FIG.  9. 


weigh    the    displaced 
wa+er. 

c.  Suspend  the  stone 
from  the  hook  on  the 
under  side  of  the 
higher  scale  pan,  and 
let  it  hang  entirely 
immersed  in  the  jar  of 
water.  Be  careful  to 
keep  it  free  from  the 
sides  and  bottom  of 
the  vessel.  (If  a  plat- 
form balance  is  used, 
Weigh  it  thus.  This  is  called 


adjust  as  shown  in  Fig.  9.) 
the  weight  of  the  stone  in  water. 

d.  Weigh  the  stone  again,  entirely  immersed  in  water  as 
before,  but  with  a  greater  or  less  depth  of  water  above  it  than 
before. 

After  the  stone  is  wholly  immersed,  does  further  lowering 
affect  its  weight  in  water  ?  Explain. 


FORM  OF  RECORD 


a.  Weight  of  stone  in  air 

b.  Weight  of  beaker 


g. 
g. 


Weight  of  beaker  and  displaced  water  —  --  g. 
Weight  of  water  displaced  —  g. 

c.  Weight  of  stone  in  water  =  -  g. 

d.  Weight  of  stone  with  a  greater  depth 

of  water  above  it  -  g. 

Loss  of  weight  of  stone  in  water  =  -  g. 


BUOYANCY   OF    LIQUIDS  83 

e.   Account  for  the  loss  of  weight  of  the  stone  in  water. 

When  the  stone  hangs  suspended  by  the  thread  in  water, 
what  forces  sustain  its  whole  or  true  weight  ? 

/.  Your  results  should  show  (within  1%  of  error)  a  simple 
relation  between  the  weight  of  the  displaced  water  and  the 
loss  of  weight  of  the  stone.  State  the  relation  and  compute 
the  per  cent  of  error. 

II.  To  find  the  relation  between  the  loss  of  weight  of 
the  stone  in  a  solution  of  table  salt  and  the  weight  of 
the  solution  that  it  displaces. 

Repeat  the  above  work  (omitting  paragraph  d)  with  the 
solution  of  table  salt.  When  you  have  finished,  return  the  solu- 
tion to  the  proper  jar. 

Compare  the  results  with  those  of  Part  I.  In  what  respects 
are  they  alike  ? 

III.  To  find  the  relation  between  the  weight  of  a  blook 
of  wood  and  the  weight  of  water  that  it  displaces  when 
floating. 

a.  Weigh  the  block  of  wood. 

b.  Determine  with  the  overflow  can  and  scales  the  weight  of 
the  water  that  the  floating  block  displaces.     Return  the  water 
to  the  proper  jar,  and  leave  the  can  and  beaker  empty  and  the 
table  dry. 

c.  What  force  or  forces  sustain  the  weight  of  the  floating 
block? 

State  and  account  for  the  resemblances  and  differences  between 
these  results  and  those  of  Parts  I  and  II. 

Discussion.  —  a.  Assuming  that  the  relation  that  is  seen  to 
hold  in  the  experiments  of  this  exercise  is  true  for  all  solids  in 
all  liquids,  how  would  you  state  it,  —  (1)  for  immersed  bodies  ; 
(2)  for  floating  bodies  ? 

b.   Account  for  the  buoyancy  of  liquids. 
COLEMAN'S  PHY.  LAB.  MAN.  —  3 


34  MECHANICS   OF    FLUIDS 

EXERCISE   62.     THE  BUOYANT  FORCE  OF  WATER 
(ALTERNATIVE   WITH   EXERCISE  61) 

References.  —  Hoadley,  143-144 ;  Carliart  and  Chute,  134- 
138. 

Apparatus.  —  Specific  gravity  balance  and  weights ;  brass 
cylinder  and  bucket ;  tumbler  or  jar  of  water ;  beaker ;  ring 
stand ;  dropping  tube. 

I.  To  find  the  relation  between  the  loss  of  weight  of  a 
brass  cylinder  in  water  and  the  weight  of  the  water 
that  it  displaces. 

a.  Observe  whether  the  cylinder  fills  the  bucket.     How  do 
the  volume  of   the  cylinder  and   the  capacity  of  the  bucket 
compare  ? 

b.  Attach  the  cylinder  to  the  hook  on  the  bottom  of  the 
bucket,  and  attach  the  bucket  to  the  hook  on  the  under  side  of 
the  higher  scale  pan.     Weigh  the  cylinder  and  bucket  in  air. 

c.  With  the  balancing  weights  still  in  the  pan,  lower  the 
cylinder  into  the  tumbler  (or  jar)  of  water  by  adjusting  the 
standard  of  the  balance.    If  the  balance  is  not  adjustable,  raise 
the  tumbler  on  blocks  or  other  supports. 

Why  is  equilibrium  destroyed  ? 

d.  With  the  cylinder  still  in  the  water, 
fill  the  bucket  exactly  level  full  of  water, 
by  means  of  the  beaker  and  the  drop- 
ping tube.  Use  the  dropping  tube  for 
adding  or  removing  a  few  drops.  Adjust 
the  balance  till  the  beam  of  the  balance 
is  horizontal  when  the  top  of  the  cylin- 
der is  just  submerged.  With  this  ad- 
justment equilibrium  should  be  restored. 
Explain  the  restoration  of  equilibrium. 

e.   State  the  principle  illustrated  by  the  experiment. 


SPECIFIC   GRAVITY   OF   SOLIDS  35 

II.  To  find  what  becomes  of  the  weight  lost  by  the 
cylinder  in  water. 

a.  Place  a  beaker  on  one  pan  of  the  balance,  with  enough 
water  in  it  to  cover  the  cylinder  completely  when  this  is  placed 
in  it ;  and  place  the  empty  bucket  in  the  other  pan.     Without 
the  cylinder  in  the  beaker,  add  weights  to  the  lighter  side  till 
balance  is  secured. 

b.  Suspend  the  cylinder  from  the  ring  stand  by  means  of  a 
thread,  so  that  it  hangs  immersed  in  the  water  of  the  beaker 
(Fig.  10).     Why  does  immersing  the  cylinder  destroy  equilib- 
rium? 

c.  Fill  the  bucket  in  the  opposite  pan  with  water.     When 
it  is  level  full  equilibrium  should  be  restored.     Why  ? 

What  forces  are  sustaining  the  whole  or  true  weight  of  the 
cylinder  ? 

EXERCISE  7.     SPECIFIC   GRAVITY   OF   SOLIDS 
References.  —  Hoadley,  46;  Carhart  and  Chute,  140-141. 

Apparatus.  —  Specific  gravity  balance  and  weights  to  1  eg., 
or  a  platform  balance  and  support  (Fig.  9),  or  a  250-g. 
spring  balance  ;  thread  ;  tumbler  or  jar  of  water ;  solid  denser 
and  one  less  dense  than  water  ;  sinker ;  mop  cloth. 

[Fairly  satisfactory  results  can  be  obtained  with  a  250-g.  (or  8-oz.) 
spring  balance,  if  the  objects  are  of  such  weight  and  density  as  to  make 
the  readings  of  the  balance  as  large  as  may  be.] 

I.  To  find  the  specific  gravity  of  a  solid  from  its  loss 
of  weight  in  water. 

Weigh  the  solid  in  air.  Suspend  it  from  the  hook  on  the 
under  side  of  the  higher  scale  pan  by  means  of  a  thread,  of 
such  length  that  it  will  be  entirely  immersed  when  the  tumbler 
(or  jar)  of  water  is  placed  beneath  the  pan.  (If  a  platform 
balance  is  used,  adjust  it  as  shown  in  Fig.  9.)  Weigh  the 
solid  in  water.  Use  the  name  of  the  solid  in  the  record. 


36  MECHANICS   OF  FLUIDS 

Compute  its  specific  gravity.  If  the  true  value  of  the  specific 
gravity  is  given  in  Table  I  of  the  Appendix,  compute  the  error 
and  the  per  cent  of  error  of  your  result. 

FORM  OF  RECORD 

Weight  of  solid1  in  air  = g. 

Weight  of  solid  in  water  = g. 

Weight  of  an  equal  volume  of  water  = g. 

Specific  gravity  of  the  solid  = 

True  value  of  its  specific  gravity  = 

Error  = 

Per  cent  of  error  = ofc 

II.  To  find  the  specific  gravity  of  a  solid  by  submer- 
sion with  a  sinker. 

Weigh  the  sinker  in  air  and  in  water.  Weigh  the  solid  in 
air.  Fasten  the  sinker  to  the  solid  so  that  both  will  be  en- 
tirely immersed  when  suspended  from  the  balance.  Weigh  the 
two  together  in  water. 

FORM  OF  RECORD 

Weight  of  sinker  in  air  = g. 

Weight  of  sinker  in  water  = g. 

Weight  of  solid1  in  air  = g. 

Weight  of  solid  and  sinker  in  water  = g. 

COMPUTATIONS 

Weight  of  water  equal  in  volume  to  solid  and  sinker  = —  —  g. 

Weight  of  water  equal  in  volume  to  sinker  = g. 

Weight  of  water  equal  in  volume  to  solid  -  g. 

Specific  gravity  of  the  solid  = — 
True  value  of  its  specific  gravity 

Error  =  — 

Per  cent  of  error  = % 

1  Use  the  name  of  the  solid  in  your  record. 


SPECIFIC   GRAVITY   OF   LIQUIDS  87 

EXEECISE   8.     SPECIFIC  GEAVITY  OF  LIQUIDS 

References.  —  Hoadley,  144-146  and  150  (c)  ;  Car  hart  and 
Chute,  144. 

Apparatus.  —  Demonstration  hydrometer  (wood  prism  of  1 
scm.  cross  section  and  graduated  in  centimeters)  ;  common 
hydrometer  (specific  gravity  scale)  for  light  and  heavy  liquids 
or  one  for  each;  hydrometer  jars  containing  water,  saturated 
solution  of  table  salt,  kerosene,  alcohol,  or  other  liquids ;  jar  of 
rinse  water ;  mop  cloth. 

I.  To  find  the  specific  gravity  of  a  liquid  by  meas- 
uring the  depth  to  which  a  woomn  prism  sinks  in  it 
and  in  water.       j, 
.  -(   - 

a.  Float  the  demonstration  hydrometer  in  water  and  meas- 
ure the  depth  to  which  it  sinks.     Record  each  item  on  a  sepa- 
rate line.    The  cross  section  of  the  hydrometer  is  1  scm.    What 
is  the  volume  ( F)  of  the  displaced  water  ?     How  does  the 
weight  of  the  water  displaced  compare  with  the  weight  of  the 
hydrometer  ? 

b.  Measure  the  depth  to  which  the  hydrometer  sinks  in  the 
salt  solution.    What  is  the  volume  (v)  of  the  displaced  solution  ? 

c.  How  does  the  weight  of  the  displaced  solution  compare 
with  the  weight  of  the  displaced  water  ?     Why  ?     Why  is  v 
less  than  V  ? 

d.  Let  w  denote  the  weight  of  the  hydrometer,  D  the  density 
of  water,  and  d  the  density  of  the  salt  solution.     The  correct 
answers  to  the  above  questions  will  give  the  relations :  w  = 
VD  —  vd.     Show  that  this  is  so. 

e.  From  the  above  equation  derive  the  proportion  d  :  D  : :  V: 
v.     This  proportion  expresses  the  relation  between  the  volume 
of  the  liquid  displaced  by  a  floating  body  and  the  density  of  the 
liquid.     Express  the  relation  in  words.     This  is  the  principle 
upon  which  the  use  of  the  hydrometer  depends. 


38  MECHANICS   OF   FLUIDS 

/.  By  definition,  the  specific  gravity  of  the  salt  solution  is 
d  -T-  D.  The  densities  are  not  measured  in  the  experiment,  but 
the  volumes  V  and  v  are;  and  from  the  above  proportion 
d  -f-  D  =  V  H-  v.  Compute  the  specific  gravity  of  the  solu- 
tion. 

I.I.  To  find  the  specific  gravity  of  liquids  by  means  of 
a  common  hydrometer. 

a.  Observe  whether  the  hydrometer  sinks  to  the  mark  1 
(or  1000)  in  water.     Does  the  reading  of  the  scale   increase 
toward  the  top  or  bottom  ?     Why  must  it  be  so  ? 

b.  Determine  the  specific  gravity  of  the  liquids  provided, 
including  the  salt  solution.     Wipe  the  hydrometer  before  put- 
ting it  from  one  liquid  into  another  and  before  putting  it  away. 
Kinse  it  in  the  jaj  of  rinse  water  after  using  it  in  the  salt 
solution. 

Discussion.  —  a.  What  changes,  if  any,  would  be  necessary 
in  the  experiment  of  Part  I  and  in  the  discussion  of  it,  if 
English  units  were  used  instead  of  metric  ?  Would  the  specific 
gravity  be  different  ? 

b.  The  specific  gravity  of  ice  is  .917.  What  fraction  of  an 
iceberg  is  above  water  ? 

EXERCISE  9.     SPECIFIC  GEAVITY  OF  LIQUIDS 

I.  To  find  the  specific  gravity  of  a  liquid  by  weighing 
a  solid  in  it  and  in  water. 

Apparatus.  —  Specific  gravity  balance  or  platform  balance 
and  support  (Fig.  9) ;  weights ;  tumbler  of  water ;  tumbler  of 
the  liquid;  a  solid  denser  than  water  or  the  liquid,  and  in- 
soluble in  them  (a  glass  stopper  is  convenient) ;  mop  cloth. 

Make  the  measurements  called  for  in  the  record,  and  com- 
pute the  specific  gravity  of  the  liquid. 


SPECIFIC    GRAVITY    OF   LIQUIDS 


39 


FORM  OF  RECORD 

Weight  of  solid 1  in  air 
Weight  of  solid  in  water 
Weight  of  solid  in  the  liquid 1 

COMPUTATIONS 


Weight  of  water  equal  in  volume  to  solid  =  —  —  g. 
Weight  of  liquid  equal  in  volume  to  solid  =  —  —  g. 
Specific  gravity  of  the  liquid 

II.  To  find  the  specific  gravity  of  a  liquid  by  balancing 
columns. 

Apparatus.  —  A  U-tube  with  water  and  kerosene  or  other 
liquid  that  will  not  mix  with  water  (Fig.  11) ;  meter  stick,  or 
metric  scale  on  the  support  of  the  tube. 

a.  Measure  the  height  of  the  liquid  col- 
umns above  the  level  (L)  of  the  surface 
separating    them    (Fig.    11).      The    liquid 
below  that  level  serves  as  a  support  for  the 
two  columns  ;  and,  since  it  is  free  to  move, 
it  would  immediately  respond  to  any  differ- 
ence in  the  pressures  exerted  upon  it  by 
these  columns.     Since  the  liquid  is  at  rest, 
these  pressures  are  evidently  equal. 

b.  Let  D  denote  the  density  of  water,  d 
the  density  of  the  liquid,  JT  the  height  of  the 
water  column  above  the  level  Z/,  and  h  the 
height  of  the  column  of  the  liquid.     Then 

the  pressures  (per  square  centimeter)  at  the  bottom  of  these 
columns  are  DH  and  dh  respectively.  Why  ?  From  this  derive 
the  formula  and  compute  the  specific  gravity  of  the  liquid. 
Remember  that,  by  definition,  specific  gravity  is  d  -+-  D. 

1  Use  the  name  of  the  solid  and  the  liquid  in  your  record. 


FIG.  11. 


40 


MECHANICS   OF   FLUIDS 


EXERCISE   10.     GAS  PRESSURE 

References.  —  Hoadley,  151-161 ;  Carhart  and  Chute,  145-150. 

I.  To  study  the  transmission  of  pressure  by  means  of 
a  Cartesian  diver. 

Apparatus.  —  An  hydrometer  jar  nearly  full  of  water,  in 
which  is  floated  a  short  glass  tube  with  bulb  blown 
at  one  end,  inverted  and  containing  just  enough 
air  to  float  it  (the  air  must  only  partly  fill  the 
tube)  ;  sheet  rubber  tied  air-tight  over  the  jar. 

a.  Press  the  fingers  on  the  rubber  cover  of  the 
jar,  and  increase  the  pressure  till  the  floating  tube 
sinks.     Diminish  the  pressure  till  it  again  rises. 
Repeat,  and  look  for  any  change  of  level  of  water 
in    the '  tube    during   the   process.     Describe   and 
account  for  this  change. 

b.  What  must  be  true  of  the  average  density  of 
bodies  that  float  ?     Of  bodies  that  sink  ?     Explain 

FIG.  12.      the  sinking  and  rising  of  the  tube. 

II.  To  study  atmospheric  pressure. 

Apparatus.  —  Bottle;  battery  jar  of  water;  glass  tube. 

a.  Fill  the  bottle  level  full  of  water  and  cover  it  with  a 
small  piece  of  paper,  being  careful  to  exclude  air   bubbles. 
Press  the  paper  smoothly  against  the  mouth  of  the  bottle,  then 
remove  the  hand  and  invert  the  bottle.     The  paper  should 
remain  in  place.     Tarn  the  bottle  in  different  directions.    With 
the  bottle  inverted,  observe   the  shape  of   the  paper.     Does 
it  bulge  as  you  would  expect  it  to  if  it  were  sustaining  the 
weight  of  the  water  ?     Explain  all  that  you  have  observed. 

b.  Hold  a  finger  over  an  end  of  the  glass  tube  and  fill  it  with 
water.     Invert  it  and  insert  the  lower  end  into  the  jar  of  water. 
Why  does  the  water  remain  in  the  tube  ?     Remove  the  finger 
from  the  top.     Why  does  the  water  in  the  tube  fall  ? 


GAS   PRESSURE 


41 


FIG.  13. 


III.  To  measure  the  pressure   of  the  gas  in  the  gas 
pipes. 

Apparatus.  —  A  water  and  a  mercury  manometer  (Fig.  13) ; 
rubber  tube. 

a.  By  means  of  the  rubber  tube  connect 
the  water  manometer  with  the  gas  jet  and 
turn  on  the  gas.     Describe  and  explain 
the  effect  upon  the  water  in  the  manom- 
eter.    Measure  the  difference  of  level  of 
the  tops  of  the  water  columns  in  the  two 
arms  of  the  manometer. 

b.  How  does  the  pressure  at  the  top  of 
the  lower  column  (the  pressure  of  the  gas) 
compare  with  the   pressure  at   the   same 
level  in  the  opposite  column?      To  what 
two  causes  is  the  latter  pressure  due  ? 

Show  that  the  total  pressure  of  the  gas 
upon  the  water  exceeds  that  of  the  air  by  an  amount  equal  to 
the  weight  of  the  water  column  sustained  by  the  gas. 

By  how  many  grams  per  square  centimeter  does  the  pressure 
of  the  gas  exceed  that  of  the  air  ? 

c.  Repeat  the  experiment,  using   the   mercury  manometer. 
Why  is  the  difference  of  level  so  much  less  than  before  ? 
Leave  the  gas  turned  off. 

IV.  To  study  the  barometer. 

Apparatus. — Two  barometer  tubes  of  different  diameters, 
filled  and  supported  vertically  by  ring  stand  and  clamp. 

a.  What  occupies  the  space  above  the  mercury  in  the 
barometer  ? 

What  is  the  pressure  upon  the  top  of  the  mercury  column  of 
the  barometer  ? 

What  measures  the  transmitted  pressure  of  the  air  exerted 
upward  in  the  tube  at  the  bottom  of  the  mercury  column  ? 


42  MECHANICS   OF   FLUIDS 

b.  Measure  the  heights  of  the  barometers.     The  height  is 
measured  from  the  surface  of  the  mercury  in  which  the  tube 
stands  to  the  top  of  the  mercury  column. 

How  is  the  height  of  a  barometer  affected  by  its  diameter  ? 

c.  Compute  the   pressure  in  grams  per  square   centimeter 
indicated  by  a  barometer  column  of  76  cm.,  taking  13.596  g. 
per  cubic  centimeter  as  the  density  of  mercury. 

EXEECISE   11.     SPECIFIC   GEAVITY   OF   A   LIQUID 

To  find  the  specific  gravity  of  a  liquid 
by  means  of  the  inverted  \J-tube. 

Apparatus.  —  Hare's  apparatus,  as  shown  in 
Fig.  14;  water  in  one  tumbler,  and  the  liquid 
whose  specific  gravity  is  to  be  determined  in  the 
other";  meter  rod. 

a.  Open  the  pinchcock,  apply  the  mouth  to 
the  tube,  and  slowly  exhaust  the  air,  while  watch- 
ing the  ascent  of  the  liquids.     Be  careful  not 
to  carry  the  exhaustion  so  far  as  to  cause  the 
higher  column  to  overflow  into  the  other.     When 
the  higher  column  is  within  a  few  centimeters 
of  the  top,  close  the  pinchcock. 

b.  If  the  tubes  were  long  enough  and  the  air 
completely  removed  from  them,  how  high  (ap- 
proximately) would   the  water   column   stand  ? 
What  would  be  the  appropriate  name  for  it  ? 

c.  Does  the  air  remaining  in  the  tube  exert 
the  same  or  different  pressure  upon  the  tops  of 

FIG.  14.          tne  two  coiunms  9 

What  balances  the  transmitted  pressure  of  the  outside  air  at 
the  bottom  of  the  two  columns  ?  (The  bottom  of  the  column  is 
at  the  level  of  the  surface  of  the  liquid  in  the  tumbler.) 

d.  Let  D  denote  the  density  of  water,  and  H  the  height  of 
the  water  column ;  d  the  density  of  the  liquid,  and  h  the  height 


C_ 


SPECIFIC   GRAVITY   OF  A   LIQUID  43 

of  its  column.  The  correct  answer  to  the  above  questions 
should  lead  easily  to  the  proof  that  DH  is  equal  to  dh.  Prove 
it,  and  from  it  derive  the  formula  for  the  specific  gravity  of  the 
liquid  in  terms  of  the  heights  of  the  columns. 

e.  Measure  the  heights  of  the  columns.  Lower  the  columns 
slightly  by  admitting  a  little  air,  and  measure  them  again. 
Again  lower  the  columns  slightly  and  measure. 

/.  Open  the  pinchcock,  and  measure  the  heights  of  the 
liquids  in  the  tubes  due  to  capillarity.  If  capillarity  causes 
depression  of  the  liquid  (as  with  mercury),  the  height  is  to  be 
recorded  as  negative.  For  computing  the  specific  gravity  of 
the  liquid,  the  heights  of  the  columns  as  they  would  have  been 
if  there  were  no  capillary  action  are  wanted.  These  corrected 
heights  are  found  by  subtracting  from  the  measured  heights 
the  elevations  due  to  capillarity  and  adding  the  depressions. 

Compute  the  specific  gravity  from  each  of  the  pairs  of 
values,  and  take  their  average. 

FORM  OF  RECORD 

123 

e.    Height  of  water  column        = cm. cm. cm. 

Height  of  liquid l  column     = cm. cm. cm. 

/.   Height   of    water    due    to 

capillarity  —  cm. 

Height   of    liquid   due    to 

capillarity  —  cm. 

Corrected  height  of  water 

column  = cm. cm. cm. 

Corrected  height  of  liquid 

column  = cm. cm. cm. 

Specific     gravity     of     the 

liquid  = 

Av.  specific  gravity  of  the 

liquid  = 

1  Use  the  name  of  the  liquid  in  your  record. 


44  MECHANICS   OF   FLUIDS 

EXERCISE   12.     THE   SIPHON   AND   THE 
SUCTION   PUMP 

References.  —  Hoadley,  169-170,  176-177 ;  Carhart  and 
Chute,  157-160. 

I.   To  study  the  action  of  a  siphon. 

Apparatus Siphon  made  of  glass  tubing,  with  arms  of  un- 
equal length  and  at  an  angle  of  about  50°  to  each  other ;  two 
battery  jars  and  water,  or,  better,  one  jar,  and  sink  at  which  to 
work ;  small  beaker ;  mop  cloth. 

a.  Invert  the  siphon  and  fill  it  from  the  beaker  by  pouring 
into  the  long  arm  and  closing  the  end  of  the  short  arm  with  a 
finger  after  it  is  filled.     When  both  arms  are  full,  stop  each 
end  with  a  finger,  and  turn  the  siphon  right  side  up  (that  is, 
with  the  bend  at  the  top).     Observe  the  effect  of  removing  the 
finger  from  either  end,  leaving  the  other  end  closed.     State  and 
account  for  the  result. 

b.  Hold  the  ends  of  the  siphon  over  the  jars  and  remove 
the  fingers  from  both  ends.     From  which  end  (the  higher  or 
the  lower)  does  the  water  run  ?  Eepeat,  and  observe  whether  the 
order  in  which  the  fingers  are  removed  affects  the  result. 

What  determines  which  way  the  water  runs  out  ?     Explain. 
Why  does  the  water  not  part  at  the  top  and  fall  from  both 
ends? 

c.  Put  water  in  the  jar  and  place  it  on  some  support  above 
the  other  jar.     Fill  the  siphon,  and,  with  a  finger  closing  the 
long  arm,  insert  the  short  arm  into  the  water  of  the  higher  jar. 
Remove  the  finger,  and  siphon  the  water  into  the  other  jar. 

Note  the  effect  of  tilting  the  siphon  so  as  to  vary  the  dis- 
tance of  the  end  of  the  long  arm  below  the  level  of  the  water 
surface.  Try  the  effect  of  raising  it  up  to  and  above  the  level 
of  the  water. 

State  and  explain  the  observed  effect. 

d.  What  has  air  pressure  to  do  with  the  action  of  a  siphon  ? 


THE   LAW   OF   BOYLE  45 

II.   To  study  the  action  of  a  suction  pump. 

Apparatus.  —  A  glass  suction  puinp;  battery  jar  of  water; 
mop  cloth. 

NOTE. — This  pump  is  called  a  lifting  pump  in  some  text-books,  in 
order  to  avoid  the  misconception  that  commonly  attaches  to  the  word  suc- 
tion. This  is  that  suction  is  a  drawing  or  pulling  process  (that  is,  one  in- 
volving a  pulling  force).  This  is  never  the  case.  Suction  is  always  a 
process  by  means  of  which  a  pressure  or  push  is  utilized.  With  a  correct 
conception  of  what  the  word  means,  there  is  no  objection  to  its  applica- 
tion to  this  kind  of  pump. 

a.  Beginning  with  the  pump  empty,  push  the  piston  down, 
while   observing   the   behavior  of    the   valve    in  the    piston. 
Explain  it. 

Lower  the  pump  into  the  jar  of  water  and  raise  the  piston 
slowly,  while  observing  the  two  valves  and  the  water  in  the 
pump.  Describe  and  explain  what  happens.  Avoid  the  use  of 
ambiguous  words,  such  as  draw,  suck,  and  suction.  State  clearly 
what  the  force  is  that  makes  the  water  rise,  and  how  it  acts. 

b.  Operate  the  pump  till  you  understand  the  motion  and  use 
of  the  valves.     Draw  figures  showing  their  action  when  the 
piston  is  ascending  and  when  it  is  descending,  and  write  a 
brief  description  of  the  action. 

EXEECISE   13.     THE   LAW  OF  BOYLE 

References.  — Hoadley,  166;  Carhart  and  Chute,  161-163. 

To  find  the  relation  between  the  volume  of  a  confined 
body  of  air  and  the  pressure  exerted  upon  it. 

Apparatus.  —  A  Boyle's-law  apparatus  with  adjustable  closed 
and  open  tubes  (Fig.  15). 

I.  a.  Adjust  the  open  and  closed  tubes  so  that  the  mercury 
stands  approximately  at  the  same  level  in  both.  What  evi- 
dence is  there  that  the  closed  tube  contains  a  gas  above  the 
mercury  ?  It  is  air. 


46 


MECHANICS   OF   FLUIDS 


How  would  the  mercury  stand  in  the  closed  tube  if  the  air 
were  removed? 

How  would  it  then  differ  from  a  barometer  ? 

b.  From  the  fact  that  the  mercury  stands  at 
the  same  level  in  the  two  arms,  what  do  you 
infer  concerning  the  relative  value  of  the  air 
pressure  upon  the  two  mercury  surfaces  ? 

The  pressure  of  the  outside  air  is  given  by 
the  barometer;  read  and  record  it  for  use 
later,  and  call  it  H. 

What  is  the  pressure  of  the  air  in  the  closed 
tube,  measured  in  centimeters  of  mercury  ? 

c.  Raise  the  open  tube  20  to  30  cm. ;  and 
while  doing  so,  observe  the  change  of  level 
in  the  closed  tube.     Has  the  confined  air  in- 
creased or  diminished  in  volume  ?     Has  the 
pressure  upon  it  been  increased  or  decreased, 
and  from  what  cause? 

The  pressure  upon  the  confined  air  is  now 
the  transmitted  pressure  of  the  outside  air 
(H)  plus  the  pressure  due  to  the  difference  of 
level  of  the  mercury  columns.  Let  d  denote 
this  difference  of  level.  This  added  pressure 
may  be  expressed  as  the  pressure  of  d  cm.  of 
mercury.  Hence  the  total  pressure  sustained 
by  the  confined  air  is  measured  by  (H+  d)  cm.  of  mercury. 

d.  Lower  the  open  tube  about  50  cm.,  while  watching  the 
change  of  level  in  the  other.  How  is  the  volume  changing  ? 
Why  is  it  changing  ? 

Again  letting  d  denote  the  difference  of  level  of  the  mercury 

columns,  what  is  now  the  pressure  upon  the  confined  air  ? 

4 

II.  a.  Clamp  the  closed  tube  near  the  bottom  of  the  stand- 
ard and  leave  it  in  this  position  for  the  first  three  sets  of  read- 
ings. Adjust  the  open  tube  so  that  the  mercury  stands  at  the 


FIG.  15. 


THE   LAW   OF   BOYLE 


47 


same  level  in  both  tubes.  Take  a  set  of  readings  as  indicated 
in  the  form  of  record.  "  Reading  of  top  of  closed  tube  "  means 
the  reading  of  the  point  on  the  meter  rod  on  a  level  with  the 
top  of  the  bore  or  hole  of  the  closed  tube.  Since  the  closed 
tube  is  of  uniform  bore,  the  volume  of  any  portion  of  it  is  pro- 
portional to  the  length  of  that  portion ;  hence  the  length  of 
the  air  column  may  be  taken  to  represent  its  volume. 

b.  Raise  the  open  tube  and  clamp  it  at  about  the  middle  of 
the  standard.     Take  a  second  set  of  readings. 

c.  Again  raise  the  open  tube  and  clamp  it  near  the  top. 
Take  a  third  set  of  readings. 

d.  Eaise  the  closed  tube  and  clamp  it  with  its  upper  end 
near  the  top  of  the  standard.     Lower  the  open  tube  and  clamp 
it  at  about  the  middle  of  the  standard.     Take  a  set  of  readings. 

e.  Lower  the  open  tube   and   clamp   it   near   the   bottom. 
Take  a  set  of  readings. 

Leave  the  tubes  clamped  at  about  the  same  height,  and  in 
such  a  position  that  the  rubber  tube  is  supported  by  the  base. 


FORM  OF  RECORD  FOR  PART  II 


SET 

READING  OF 
TOP  OP  CLOSED  TUBE 

READING  OP  TOP  OF  MERCURY  COLUMN 

In  closed  tube 

In  open  tube 

a. 

cm. 

cm. 

cm. 

b. 



-^  — 



etc. 







COMPUTATIONS 


SET 

LENGTH  OF 
AIR  COLUMN  OR  V 

DIFFERENCE  OF  LEVEL 
OF  MERCURY  COLS.  =  d 

TOTAL  PRESSURE 
=  fI±d=P 

PV 

a 

cm. 

cm. 

cm.  (of  mer  .  ) 



b 









etc. 









48  MECHANICS   OF   FLUIDS 

Discussion. — a.  According  to  Boyle's  law,  what  relation 
should  exist  among  the  numbers  of  the  column  headed  PV? 

b.  Compute  the  greatest  per  cent  of  difference  between  these 
numbers.     This  per  cent  represents  the  error,  and  should  not 
exceed  2%.     If  it  exceeds  4%  repeat  the  experiment. 

c.  Is  it  necessary  that  the  open  and  the  closed  tubes  should 
be  of  the  same  diameter  ?     Explain. 

d.  Would  the  accuracy  of  the  result  be  affected  by  any  un- 
evenness  in  the  diameter  of  the  open  tube  ?     In  the  diameter 
of  the  closed  tube  ?     Give  reasons  in  each  case. 

e.  When  the  mercury  stands  at  the  same  level  in  the  two 
arms  of  a  Boyle's-law  apparatus,  the  air  column  is  20  cm.  long. 
What  will  be  its  length  (a)  when  the  mercury  stands  25  cm. 
higher  in  the  closed  than  in  the  open  tube ;  and  (b)  when  it 
stands  30  cm.  lower ;  the  atmospheric  pressure  being  75  cm.  ? 

EXEKCISE   14.     THE   DENSITY   OF  AIR 

Reference.  —  Hoadley,  155. 

To  find  the  weight  of  one  cubic  centimeter  of  air  at 
the  temperature  and  pressure  of  the  laboratory. 

Apparatus.  —  Air  pump;  closed  mercury  pressure  gauge; 
strong  balance,  sensitive  to  1  eg. ;  weights  to  1  eg. ;  forceps ; 
heavy  rubber  tubing  for  connections ;  brass  cylinder  with  stop- 
cock at  each  end  (or  bottle  with  rubber  stopper,  in  which  is 
fitted  a  short  piece  of  glass  tubing  with  Y-tube,  pinchcock,  and 
connections  as  shown  in  Fig.  16). 

[The  cylinder  should  be  strengthened  with  braces  or  partitions  (not 
air-tight),  and  should  have  a  capacity  of  1500  to  2000  ccm.  The  capacity 
should  be  given  the  students.  A  bottle  holding  3  to  5  pts.  will  serve  instead 
of  the  cylinder.  The  pressure  gauge  has  no  air  in  the  closed  tube.] 

a.  The  capacity  of  the  cylinder  (or  bottle)  is  marked  on  it. 
Record  it. 

b.  Attach  the  cylinder  by  a  rubber  tube  to  the  air  pump, 
and  by  another  to  the  pressure  gauge  (Fig.  16) ;  and  exhaust 


THE   DENSITY   Off   AIR 


FIG.  10. 


the  air  till  the  mercury  falls  a  few  centimeters  in  the  gauge, 
or  until  the  difference  of  level  of  the  mercury  column  is  not 
more  than  10  or  15  cm.  The 
gauge  acts  as  a  short  barometer 
to  measure  the  greatly  reduced 
pressure  in  the  cylinder.  Quickly 
read  the  heights  of  the  mercury 
columns  of  the  gauge;  then  close 
the  stopcocks.  Very  slowly  re- 
move the  tube  connecting  with  the 
gauge.  If  it  is  removed  quickly, 
the  sudden  inrush  of  air  may  make 
the  mercury  strike  the  closed  end  with  sufficient  force  to  break 
it.  Disconnect  the  other  tube  and  weigh  the  cylinder.  If  the 
balance  is  provided  with  a  lever  or  other  device  for  raising 
and  lowering  the  beam  and  pans,  always  lower  the  beam  before 
placing  anything  on  or  removing  anything  from  the  pans,  except 
weights  lighter  than  1  g.  See  that  the  pans  are  suspended 
from  the  knife  edges  on  the  beam.  They  sometimes  slip  off. 

c.  Open  the  stopcocks  and  weigh  again.     Kecord  each  item 
on  a  separate  line.     The  difference  between  the  two  weighings 
is  the  weight  of  the  air  pumped  out.     Compute  it. 

d.  Eead  the  barometer.     At  the  second  weighing,  the  air  in 
the  cylinder  was  under  this  pressure ;  at  the  first  weighing,  it 
was  under  the  pressure  measured  by  the  difference  of  level  of 
the  mercury  columns  of  the  gauge.     The  weight  of  air  in  a 
given  space  and  at  a  given  temperature  is  proportional  to  its 
pressure.     (This  follows  from  Boyle's  law.) 

e.  From  the  known  pressures,  find  what  (decimal)  fraction 
of  the  air  was  pumped  out. 

/.  Compute  the  weight  of  the  air  in  the  cylinder  when  under 
atmospheric  pressure. 

g.  Compute  the  density  of  the  air  (in  grams  per  cubic 
centimeter)  at  the  temperature  and  pressure  of  the  laboratory. 


COLEMAN'S  PHY.  LAB.  MAN. — 4 


OF  THE 


IV.     MECHANICS    OF   SOLIDS 
FORCES  i 

27.  A  force  is  a  push  or  a  putt,  and  tends  to  cause  motion  of 
the  body  on  which  it  acts  or  to  change  the  existing  motion  of 
that  body.     It  will  do  so  unless  prevented  by  the  equal  and 
opposite  tendency  of  another  force  or  other  forces. 

28.  Balanced  Forces.     Equilibrium.  —  If  two  or  more  forces 
act  upon  a  body  at  rest  so  as  completely  to  neutralize  each 
other's  tendency  to  produce  motion,  the  body  will  remain  at 
rest,  and  is  said  to  be  in  equilibrium.     The  forces  also  are  said 
to  be  in  equilibrium  or  to  balance  each  other ;  and  are  often 
called  balanced  forces.    A  large  part  of  the  work  of  this  chapter 
consists  in  a  study  of  the  conditions  of  equilibrium ;  that  is,  of 
the  relations  that  must  exist  among  several  forces  in  order  that 
they  shall  balance  each  other. 

29.  The  resultant  of  two  or  more  forces  is  the  single  force 
that  would  produce  the  same  effect  as  the  given  forces,  if  it 
were  substituted  for  them.     The  given  forces  are  called  com- 
ponents (that  is,  parts)  of  the  resultant.     The  resultant  of  any 
number  of  forces  in  equilibrium  is  zero ;  for,  being  in  equilib- 
rium, they  do  not  affect  the  state  of  rest  or  of  motion  of  the 
body  upon  which  they  act. 

30.  In  any  problem  concerning  the  conditions  of  rest  or  of 
motion  of  a  body,  we  are  always  at  liberty  to  replace  given 

1  The  summary  presented  in  these  articles  is  intended  to  serve  for  con- 
venient reference  in  the  laboratory,  not  as  a  substitute  for  the  detailed 
treatment  to  be  found  in  text-books. 

50 


COMPOSITION   OF   FORCES  51 

forces  by  their  resultant  or  a  given  force  by  its  components ; 
since,  if  such  a  substitution  were  actually  made,  the  behavior 
of  the  body,  as  regards  rest  or  motion,  would  not  be  affected 
in  the  least. 

31.  The  equilibrant  of  any  number  of  forces  (one  or  more) 
is   the   single   force   that   would   hold    the    given  forces    in 
equilibrium. 

32.  The  elements  of  a  force  are, —  (1)  its  point  of  application; 
(2)  its  direction;  (3)  its  magnitude.     A  force  is  not  completely 
known  or  determined  till  these  three  things  are  known  about 
it.     A  comparison  of  two  forces  consists  in  a  comparison  of 
their  three  elements. 

33.  Representation  of  Forces.  —  A  force  may  be  represented 
in  all  its  elements  by  a  straight  line.     One  end  of  the  line 
represents  the  point  of  application  of  the  force ;  the  direction 
of  the  line  (with  an  arrow  head  on  it,  since  a  line  has  two 
directions)    represents   the   direction   of   the   force;    and   the 
length  of  the  line  represents  the  magnitude  of  the  force  on 
a  definite  scale,  arbitrarily  adopted  to  suit  convenience. 


BALANCED   FORCES:    STATICS 
EXEECISE   15.     COMPOSITION   OF  FORCES 

References.  —  Hoadley,  47-49 ;  Carhart  and  Chute,  45-47. 
I.    To  study  the  conditions  of  equilibrium  of  two  forces. 

Apparatus.  —  Three  2000-g.  spring  balances,  preferably  with 
flat  backs ;  two  nails ;  three  cords  attached  to  small  ring ;  large 
board  on  which  to  perform  the  experiment,  with  narrow  strips 
on  two  adjacent  sides,  in  which  are  bored  holes  1  in.  apart  to 
hold  the  nails  (Fig.  17)  ;  rule. 


52 


MECHANICS   OF   SOLIDS 


a.  Attach  two  of  the  spring  balances  to  cords  on  the  ring, 
and,  with  one  in  each  hand,  stretch  the  cords  and  leave  the 
ring  at  rest.     The  ring  is  now  in  equilibrium  under  the  action 
of  two  pulls. 

How   do   these    pulls    compare    in    their    three    elements 
(Art.  32)? 
Do  you  alter  these  relations  when  you  vary  the  pulls  ? 

b.  Either  of  two  forces  in  equilibrium  is  the  equilibrant  of 
the  other  (Art.  31).     What  is  the  resultant  of  the  two  ? 

II.    To  study  the  conditions  of  equilibrium  of  three 
concurrent  forces. 

a.  Attach  the  cords  on  the  ring  to  the  hooks  of  the  three 
balances.     Fasten  two  of  the  balances  by  their  rings  to  the 
nails,  inserted  in  holes  in  the  board,  and  placed  so  that  the 

angle  between  these  bal- 
ances will  be  between  60° 
and  120°  (Fig.  17).  Place 
your  record  sheet  under 
the  cords.  Pull  on  the 
ring  of  the  third  balance 
with  sufficient  force  and 
in  the  proper  direction  to 
cause  each  of  the  balances 
to  register  not  less  than 
1000  g.  and  not  more 
than  2000  g.  Adjust  the 
paper  under  the  cords, 
hold  the  third  balance 
steadily,  and  make  a  dot 
on  the  paper  at  the  middle  of  the  ring  and  one  exactly  under 
each  of  the  cords  at  a  considerable  distance  from  the  ring.  Kead 
each  of  the  balances  and  record  on  the  sheet  beside  the  cords. 

b.  Remove  the  sheet,  draw  lines  through  the  dots  indicating 
the  directions  of  the  cords  (and  hence  also  the  directions  of 


FIG.  17. 


COMPOSITION   OF   FOKCES  53 

the  forces),  and  measure  off  on  these  lines,  from  their  point 
of  intersection,  distances  proportional  to  the  forces,  using  a 
scale  that  will  give  a  rather  large  figure  (Art.  33).  The  remain- 
der of  the  construction  may  be  left  for  home  work.  Con- 
struct accurately,  by  the  parallelogram  of  forces,  the  resultant 
of  any  two  of  these  forces  ;  and  compare  it  with  the  third  force 
(Art.  32). 

The  third  force  is  the  equilibraut  of  the  other  two  (Art.  31). 
Is  it  the  equilibrant  of  their  resultant  ? 

c.  Arrange  the  balances  so  that  two  of  them  will  act  at 
an  angle  of  exactly  90°  after  the  forces  are  applied.     For  an 
angle  to  measure  by,  fold  a  piece  of  paper  twice.     Draw  a 
figure  to  scale  as  in  the  first  case,  marking  in  the  figure  the 
magnitude  of  each  force.     Record  the  scale  adopted  for  the 
figure. 

d.  Obtain  by  construction  the  resultant  of  the  two  forces 
acting  at  an  angle  of  90°,  and  mark  the  value  of  the  resultant 
in  the  figure. 

Find  the  same  resultant  by  computation.  (The  square  on 
the  hypotenuse  of  a  right  triangle  is  equal  to  the  sum  of  the 
squares  on  the  other  two  sides.) 

Find  the  per  cent  of  difference  between  these  two  results. 
The  error  should  not  exceed  2%. 

Discussion.  —  a.  In  the  above  work,  how  should  the  resultant 
of  any  two  of  the  forces  compare  with  the  third  force  ?  Give 
a  full  and  satisfactory  reason  for  your  answer. 

b.  What  is  the  resultant  of  the  three  forces  applied  to  the 
ring  ?     Give  reasons  for  your  answer. 

c.  As  the  angle  between  two  forces  increases  from  0°  to  180°, 
how  does  their  resultant  vary  ?     What  ^is  the  value  of  the 
resultant  at  the  beginning  ?     At  the  end  ? 

d.  Two  equal  forces  act  at  an  angle  of  120°.     What  is  the 
magnitude  and  direction  of    their  resultant?     Answer  from 
geometry. 


54  MECHANICS  OF   SOLIDS 

EXERCISE   16.     EQUILIBRIUM   OF  PARALLEL 
FORCES 

References.  —  Hoadley,  51 ;  Carhart  and  Chute,  47. 

To  study  the  conditions  of  equilibrium  of  three 
parallel  forces. 

Apparatus.  —  Meter  rod;  two  2000-g.  spring  balances;  two 
unequal  weights  of  2000  to  3000  g. ;  cord ;  frame  to  support 
the  balances  (Fig.  28)  ;  rule. 

a.  Adjust  the  balances  and  meter  rod  as  shown  in  Fig.  18. 
The  supporting  cords  must  be  parallel  and  the  rod  horizontal. 
For  convenience  in  reading  the  distances,  the  balances  may  be 
hung  80  cm.  apart  and  the  cords  attached  to  the  rod  10  cm. 
from  each  end.  Take  the  readings  of  the  balances  when  sup- 


FIQ.  18. 

porting  the  rod  only.  Estimate  to  tenths  of  the  smallest  scale 
division  (Art.  26).  Record  the  readings  as  the  zero  readings 
of  the  left  and  right  scales  respectively.  They  are  the  forces 
necessary  to  support  the  rod,  and  are  to  be  subtracted  from 
subsequent  readings  of  the  scales  in  order  to  obtain  the  forces 
(/  and  jP,  Fig.  19)  necessary  to  balance  the  weight  (W)  hung 


EQUILIBRIUM   OF  PARALLEL  FORCES 


55 


upon  the  rod.  In  order  to  keep  the  zero  readings  the  same 
throughout  the  exercise,  the  rod  must  always  be  supported 
from  the  same  points. 

b.  In  Fig.  19,  A  and  B  denote  the  points  of  support  of  the 
rod,  and  C  the  point  of  application  of  the  weight.      In  the 
record  let  the  distances  AC 

and  CB  be   denoted  by  d       f 
and  D  respectively.     Hang 
one  of  the  weights  on  the    - 
rod    midway   between    the 
supporting  cords,  and  record 
the  readings  of  the  scales 
and  the   equal  distances  d  w 

and  D.    Draw  a  figure  simi- 
lar to  Fig.  19,  representing 
the  distances   and  the  forces  approximately  to   scale.      (An 
accurate  construction  is  not  required.) 

c.  Move  the  weight  from  10  to  20  cm.  toward  one  end,  and 
take  a  second  set  of  readings.     Draw  a  figure  as  before. 

d.  Use  the  other  weight.     Hang  it  so  that  d  and  D  are  un- 
equal, and  take  a  third  set  of  readings.     Draw  figure  to  illus- 
trate.    When  you  have  finished,  remove  the  weight  from  the 
rod. 

FORM  OF  RECORD 


a.    Zero  reading  of  left  scale    = 
Zero  reading  of  right  scale  = 


FIG.  19. 


SCALE  I 

IEADINO 

If 

t/ 

/) 

Left 

Riffht 

J) 

g- 

%• 

—  g. 

cm. 

cm. 

d 





— 





MECHANICS   OF   SOLIDS 


COMPUTATIONS 


Cp-p 

FOKOKS 

f  •  F 

D  '  d 

DlFFKR- 

R  —  f+  F 

R      W 

/ 

F 

i  \i  i: 

b 

<v 

tv 

g. 





—  •=- 

—  % 

&' 



d 

— 















Discussion.  —  a.  Enter  in  the  record,  as  indicated,  the  com- 
puted values  (expressed  decimally)  of  the  ratios  /:  F  and  D :  d ; 
and  compute  the  per  cent  of  difference  between  them  for  each 
set  of  readings.  This  difference  is  due  to  experimental  errors, 
and  should  not  exceed  1%.  If  it  is  greater  than  2%  in  any 
case,  repeat  the  set  of  readings. 

6.  Since  the  ratios  are  equal,  they  form  a  proportion. 
Write  it,  using  the  letters  /,  F,  d,  and  D ;  arid  state  it  in  words 
also. 

c.  What  is  the  magnitude,  direction,  and  point  of  application 
of  the  equilibrant  (W)  of  two  parallel  forces  acting  in  the 
same  direction  ? 

c?.  How  should  R,  the  resultant  of  /  and  F,  compare  with 
W  in  magnitude,  direction,  and  point  of  application  ? 


ALTERNATIVE  METHOD 

Apparatus.  —  Meter  rod ;  three  2000-g.  spring  balances  ;  cord. 

Follow  the  above  directions  with  the  following  modifications  : 
A  third  spring  balance  is  used  instead  of  the  weight  W,  and 
the  rod  and  balances  lie  upon  the  table.  There  will  be  no  zero 
reading  of  the  balances.  The  reading  of  the  single  opposing 
balance  should  in  every  case  be  near  its  maximum,  in  order  to 
secure  the  best  results.  The  two  balances  acting  in  the  same 
direction  may  be  fastened  to  nails  driven  into  the  table,  and 
the  third  balance  held  in  the  hand. 


MOMENTS   OF    FORCE  57 

EXERCISE   17.     iMOME>sTTS    OF   FORCE 

Reference.  —  Hoadley,  96-98. 

To  study  the  conditions  of  equilibrium  of  two  forces 
about  an  axis. 

Apparatus.  —  Meter  rod  with  hole  at  50  cm.;  upright  with 
nail  to  support  the  rod ;  2000-g.  spring  balance ;  four  weights, 
two  of  which  are  equal ;  rule. 

[The  hole  in  the  rod  should  be  exactly  at  50  cm.  and  slightly  displaced 
laterally,  so  that  the  rod  will  balance  horizontally  with  slight  stability  on 
a  nail.  Few  meter  rods  are  uniform  enough  in  cross  section  and  density 
to  balance  exactly  at  50  cm. ;  but  this  defect  is  remedied  by  boring  small 
holes  in  the  heavier  end.] 

a.  Hang  the  rod  on  the  support.  Hang  the  equal  weights 
on  the  rod,  one  on  each  side,  and  adjust  them  so  that  the  rod 
will  balance  in  a  horizontal  position.  In  the  record  and  in  the 
figure  let  W  and  iv  denote  the  weights,  and  A  and  a  respec- 
tively their  arms  (the  dis- 
tances from  the  weights  to  — A  — 


W          FIG.  20. 


the  axis  of  rotation).  The 
axix  of  rotation  is  also  called 
the  fulcrum.  The  product 
of  either  weight  and  its  arm 
is  called  the  moment  of  the  weight  about  the  fulcrum ;  thus 
the  moment  of  the  weight  w  about  the  fulcrum  is  wa,  and 
that  of  W  is  WA.  Draw  a  figure  similar  to  Fig.  20,  approxi- 
mately to  scale,  and  in  it  record  the  values  of  the  weights  and 
their  arms. 

b.  Hang  unequal  weights  on  the  rod  and  adjust  them  so 
that  they  balance.  Better  results  will  be  obtained  by  making 
the  arms  rather  long.  Compute  the  moments  of  these  fordes. 
Within  a  very  small  error  (less  than  .5%),  a  simple  relation 
should  hold  between  them.  What  is  it?  Draw  a  figure  as 
before. 


58 


MECHANICS   OF   SOLIDS 


c.  Repeat  the  preceding  with  other  unequal  weights. 

d.  Hang  the  heaviest  weight  on  the  rod,  and  balance  by 
holding  up  the  same  end  of  the  rod  by  the  hook  of  the  spring 
balance,  applied  nearer  the  end  of  the  rod  than  the  weight. 
Let  W  denote  the  weight  and  A  its  arm  ;  and  let  /  denote  the 
force  applied  through  the  balance  and  a  its  arm.     Eemember 
that  the  arm  is  the  distance  from  the  point  of  application  of 
the  force  to  the  axis  of  rotation.     Compute  the  moments  of  W 
and  /  about  the  axis,  and  find  the  per  cent  of  difference  between 
these  moments. 

A  larger  error  may  be  expected  than  in  the  previous  work. 
Why? 

Draw  a  figure  showing  W  and  /  drawn  approximately  to 
scale  and  in  the  proper  directions. 

e.  Repeat  the  preceding  with  the  balance  nearer  the  axis 
than  the  weight  is. '   Draw  figure  as  before. 

FORM  OF  RECORD 


W 

W 

a 

A 

wa 

WA 

DlFF. 

%OFDIFF. 

o/ 

b 

— 

»• 











7o 

a 

/ 

fa 

Discussion.  —  a.  Express  algebraically,  both  as  an  equation 
and  as  a  proportion,  the  relation  that  should  hold  for  the  four 
quantities  W,  w,  a,  and  A ;  and  also  for  W,  f,  A,  and  a. 
Express  the  same  relation  in  words. 

b.  Compare  the  conditions  of  equilibrium  when  the  forces 
are  applied  on  opposite  sides  of  the  fulcrum  with  the  condi- 
tions when  they  are  applied  on  the  same  side. 


CENTER   OF   GRAVITY  59 

c.  In  Exercise  16  find  the  moments  of  the  forces  /  and  F 
about  an  axis  through  the  rod  at  C  (the  point  of  application 
of  the  weight)  for  one  set  of  observations. 

d.  Compare  Exercises  16  and  17.     In  what  respects  are  they 
alike  ?     What  and  where  is  the  equilibrant  of  w  and  W  in  this 
exercise  ? 

34.  Center  of  Gravity.  —  The  attraction  of  the  earth  is  exerted 
on  all  parts  of  a  body,  — an  exceedingly  .small  pull  upon  each 
particle.     These  pulls  are  directed  toward  the  center  of  the 
earth;   hence  they  are  practically  parallel.     The  resultant  of 
these  parallel  forces  is  equal  in  magnitude  to  their  sum ;  and 
we   call   it   the   weight   of   the   body.     (This   is   not  strictly 
accurate  except  for  bodies  at  the  poles.)     The  direction  of  this 
resultant  is  the  same  as  the  direction  of  its  components,  — 
vertically  down.      Its  point  of  application  is  named  the  center 
of  gravity  of  the  body.    Hence  we  have  the  following  definition : 

The  center  of  gravity  of  a  body  is  the  point  of  application 
of  the  resultant  of  all  the  forces  of  gravity  acting  upon  the 
body. 

35.  Properties  of  the  Center  of  Gravity.  —  The  center  of  gravity 
of  a  body  has  several  important  properties,  two  of  which  are 
fundamental,  and,  if  clearly  grasped,  will  greatly  assist  in  the 
understanding  of  all  questions  that  may  arise  under  this  topic. 
They  are  the  following :  — 

(1)  So  long  as  a  body  remains  intact  and  of  the  same  shape, 
its  center  of  gravity  is  a  fixed  point  relative  to  the  body,  how- 
ever the  body  may  be  turned  and  in  whatever  situation  it  may 
be  placed. 

(2)  The  second  property  follows  directly  from  the  definition 
of  a  resultant  force  (Art.  29).     It  is  that,  under  all  conditions 
affecting  the  state  of  rest  or  of  motion  of  a  body,  the  body 
behaves  exactly  as  it  would  if  the  actual  forces  of  gravity  (in- 
definite in  number)  were  replaced  by  a  single  force  (the  weight 
of  the  body)  applied  at  the  center  of  gravity  and  acting  verti- 


b'O  MECHANICS   OF   SOLIDS 

cally  downward.  In  problems  in  equilibrium  this  substitution 
of  the  resultant  for  the  actual  forces  is  always  assumed  at  the 
outset. 

EXERCISE   18.     CENTER  OF   GRAVITY   AND 
EQUILIBRIUM 

References.  —  Hoadley,  70, 71, 74,  and  75 ;  Carhart  and  Chute, 
51-52  and  55-58. 

I.  To  find  the  center  of  gravity  of  an  irregular  piece 
of  cardboard. 

Apparatus  (for  Parts  I  and  II).  —  Irregular  piece  of  card- 
board ;  pin ;  small  plumb  line  made  of  thread  and  bullet  or 
button;  rule. 

a.  Stick  the  pin  through  the  cardboard  near  the  edge,  and 
enlarge  the  hole  till  the  cardboard  swings  freely  on  the  pin. 
Hang  the  plumb  line  on  the  pin  near  the  cardboard  but  not 
quite  touching  it.  When  both  have  come  to  rest,  grasp  them 
together  at  the  bottom,  make  a  dot  accurately  under  the  thread, 
and  with  a  sharp  pencil  and  rule  draw  a  line  connecting  this 
dot  with  the  point  of  suspension.  Do  all  this  with  care.  Sus- 
pend the  cardboard  at  a  different  point,  also  near  the  edge, 
and  determine  a  second  line  in  the  same  way. 

6.  What  single  point  of  the  cardboard  was  vertically  under 
the  point  of  suspension  in  both  cases  ?  Do  you  think  it  would 
be  vertically  under  any  point  of  suspension  from  which  the 
cardboard  hangs  at  rest  ?  Test  the  matter  by  suspending  it 
from  a  third  point  chosen  at  random.  State  the  result. 

c.  If  the  pull  of  the  earth  upon  the  molecules  of  the  card- 
board were  replaced  by  their  resultant  (the  weight  of  the  card- 
board), where  would  it  have  to  be  applied  to  cause  the  cardboard 
to  behave  as  it  does  when  suspended  from  different  points? 
Give  reason  for  your  answer.  Trace  the  outline  of  the  card- 
board in  your  notebook,  locate  the  three  points  of  support,  and 
draw  the  plumb  lines.  Letter  the  center  of  gravity  C. 


CENTER    OF   GRAVITY   AND    EQUILIBRIUM  61 

II.  To  study  the  different   kinds  of  equilibrium  by 
means  of  the  cardboard  suspended  from  a  pin. 

a.  Hang  the  cardboard  again  from  a  hole  near  the  edge  and 
set  it  swinging.     Describe  the  path  of  the  center  of  gravity. 

Where  is  the  center  of  gravity  with  respect  to  the  lowest 
point  of  that  path  when  the  cardboard  hangs  at  rest  ? 

Is  the  center  of  gravity  raised  or  lowered  when  the  cardboard 
is  moved  from  its  position  of  equilibrium  ? 

What  name  is  given  to  this  kind  of  equilibrium  ?     Define  it. 

b.  Hang  the  cardboard  on  the  pin  through  the  center  of 
gravity;   enlarge  the  hole  till  it  moves  freely;   and  observe 
its  behavior  when  it  is  set  rotating.     Does  it  always  come  to 
rest  in  the  same  position  ?     It  is  not  easy  to  find  and  pierce 
the  center  of  gravity  exactly  with  the  pin.    It  generally  happens 
that  the  hole  is  far  enough  to  one  side  to  affect  appreciably  the 
behavior  of  the  cardboard.     Have  you  reason  to  think  that 
such  is  the  case  in  your  experiment  ?     If  so,  state  it. 

How  would  the  cardboard  behave  with  respect  to  positions 
of  rest  if  it  were  suspended  accurately  at  the  center  of  gravity  ? 

In  what  kind  of  equilibrium  would  it  be  ?  Define  this  kind 
of  equilibrium. 

c.  Suspend  the  cardboard  at  one  of  the  outer  holes,  and  try 
to  balance  it  with  the  center  of  gravity  vertically  above  the 
support.      Why  is  this  kind  of  equilibrium  difficult  to  secure  ? 
Name  and  define  this  kind  of  equilibrium. 

III.  To  study  weight  as  a  resultant  force  by  means 
of  a  stick  of  uniform  cross  section. 

Apparatus —  Meter  stick  ;  platform  balance  and  weights ; 
fulcrum  to  support  the  meter  rod. 

a.  Weigh  the  meter  rod  on  the  platform  balance,  and  let  w 
denote  its  weight.  Find  the  center  of  gravity  of  the  rod  by 
balancing  it  horizontally  upon  the  support.  Do  not  waste 
time  trying  to  secure  a  perfect  balance :  the  equilibrium  is 
unstable.  When  balanced,  the  center  of  gravity  is  above  the 


62  MECHANICS   OF   SOLIDS 

support.  Record  its  position  as  the  reading  of  the  meter  scale 
at  that  point.  On  account  of  slight  variation  in  the  density 
or  cross  section  of  the  rod,  its  center  of  gravity  may  be  appre- 
ciably to  one  side  of  50  cm. 

b.  The  purpose  of  this  experiment  is  to  determine  whether, 
under  different  conditions  of  equilibrium,  the  weight  of  the 
whole  rod  may  still  be  regarded  as  a  single  force  whose  point 
of  application  is  the  center  of  gravity  of  the  whole  rod. 

c.  Hang  a  100-g.  weight  (denoted  by  W)  1  cm.  from  the 
zero  end  of  the  rod;   then  balance  the  rod  on  the  support. 
Assuming  that  W  is  balanced  by  w  (which,  remember,  is  the 
weight  of  the  whole  rod),  we  can  find  the  point  where  w  must 
be  applied ;  for,  if  W  and  w  balance  each  other  about  the  axis 
of  support,  their  moments  about  that  axis  must  be  equal.     Let 
A  and  a  denote  the  arms  of  W  and  w,  respectively.     Measure 
A,  and  record  it  and  the  position  of  the  axis  of  support.    Com- 
pute a  from  the  equation  wa  =  WA. 

d.  We  have  now  found  that,  in  order  to  balance  W,  the 
weight  of  the  rod  must  be  applied  a  cm.  from  the  axis  of 
support.     What  is  this  position  on  the  rod?     Is  it  the  same 
point  as  the  center  of  gravity  ?     It  should  be,  within  a  reason- 
able limit  of  experimental  error  (2  or  3  mm.). 

e.  What  answer  have  you  found  for  the  question  stated  in 
paragraph  b  ?     (See  Art.  35.) 

FORM  OF  RECORD 

a.    Weight  of  meter  rod  (w) 
Position  of  center  of  gravity 

c.  Attached  weight  (W)  = g. 

Position  of  attached  weight  =  I  cm. 

Position  of  axis  of  support  = cm. 

Arm  of  W  (A)  = cm. 

Arm  of  the  weight  of  the  rod  (a)  =  WA  -5-  W    — cm. 

d.  Point  of  application  of  the  weight  of  the  rod 

—  position  of  the  axis  of  support  -f-  a          =  -    -  cm. 


EQUILIBRIUM  AND   STABILITY  63 

EXEECISE  19.     EQUILIBRIUM  AND  STABILITY 
References.  —  Hoadley,  76-77;  Carhart  and  Chute,  57-59. 

I.  To  study  the  different   kinds  of  equilibrium  by 
means  of  bodies  supported  on  a  level  surface. 

Apparatus —  Such  bodies  as  cylinder,  cone,  sphere,  oblate 
and  prolate  spheroids,  an  empty  round-bottomed  flask,  a  round- 
bottomed  flask  loaded  with  shot  so  that  it  will  stand  upright. 

[The  shot  can  be  kept  in  place  by  paraffine  or  wax  that  has  been 
melted  over  it.] 

a.  A  body  may  have  different  kinds  of  equilibrium  at  the 
same   time   with   respect    to   motion   in   different   directions. 
Experiment  with  the  different  bodies  provided,  and  determine 
their  kinds   of   equilibrium   in   different   positions  and  with 
respect  to  motion  in  different  directions   for  each   position. 
Give  a  complete  account  of  each  case  investigated,  with  draw- 
ings  to   illustrate.      Include   cases    of   unstable   equilibrium, 
whether  you  can  perfectly  realize  them  or  not.     Locate  as 
closely  as  possible  the  center  of  gravity  in  each  drawing,  and 
indicate  by  a  dotted  line  the  path  that  it  would  describe  if 
the  equilibrium  of  the  body  were  disturbed. 

b.  Balance  your  pencil  on  its  point  on  your  finger,  making 
use  of  a  pocketknife  to  secure  stable  equilibrium.     The  blade 
should  be  half  open  and  stuck  into  the  pencil  near  its  point  so 
that  the  handle  hangs  below  the  finger.     Draw  a  figure,  and  in 
it  locate  the  center  of  gravity  of  the  pencil  and  knife  regarded 
as  one  body. 

How  definitely  does  the  experiment  determine  this  center 
of  gravity  ? 

II.  To  study  the  conditions  affecting  the  stability  of 
bodies. 

Apparatus.  —  A  rectangular  and  a  trapezoidal  block  (Figs.  22 
and  21),  cut  from  a  2-in.  board,  the  former  heavily  loaded 


64  MECHANICS  OF   SOLIDS 

with  lead  at  one  end,  and  the  latter  loaded  on  the  shorter  of 
the  parallel  sides  so  as  to  bring  the  center  of  gravity  midway 
between  these  sides.  In  one  of  the  broad  sides  of  each  of  these 
blocks  drive  a  nail  over  the  center  of  gravity  for  the  suspension 
of  a  plumb  line ;  and  bore  two  holes  in  each  near  vertices  for 
suspension  in  verifying  the  location  of  the  center  of  gravity. 

a.  A  nail  is  driven  in  a  side  of  each  of  the  blocks,  presum- 
ably over  the  center  of  gravity  (which,  of  course,  is  within  the 
block).  Verify  the  location  of  the  center  of  gravity  of  each 
block  by  suspending  it  by  a  nail  or  a  short  wire  from  each  of 
the  holes  in  succession,  and  noting  the  direction  of  the  plumb 
line  suspended  from  the  same  support. 
b.  Investigate  the  degree  of  diffi- 
culty in  overturning  the  trapezoidal 
block  on  a  short  edge  from  a  position 
of  equilibrium  on  each  of  its  parallel 
bases  (Pig.  21). 

What  relation  do  you  discover  be- 
FIG  21  tween  the  area  of  the  base  and  the 

stability  of  the  body  (the  height  of 
the  center  of  gravity  remaining  the  same)  ? 

Draw  a  figure  of  the  block  in  each  position,  and  show  by  a 
dotted  line  the  path  of  the  center  of  gravity  in  the  process  of 
overturning. 

What  relation  do  you  discover  between  the  vertical  distance 
through  which  the  center  of  gravity  must  be  raised  before  the 
body  falls  over  and  the  difficulty  of  overturning  it?  (The 
nature  of  the  experiment  does  not  justify  the  statement  of 
definite  relations,  such  as  proportionality.) 

c.  Investigate  the  degree  of  difficulty  in  overturning  the 
rectangular  block  on  a  short  edge  from  a  position  of  equi- 
librium on  each  end. 

What  relation  do  you  discover  between  the  height  of  the 
center  of  gravity  of  a  body  and  its  stability  (the  area  of 
the  base  remaining  the  same)  ? 


COMPARISON   OF   MASSES    BY   INERTIA  65 

Draw  figures  as  before,  showing  the  path  of  the  center  of 
gravity  in  overturning. 

Is  the  distance  through  which  the  center  of  gravity  must  be 
raised  to  overturn  the  block  the  same  or  different  for  the  two 
ends?  What  suggestion  does  the  answer  to  this  question 
afford  in  regard  to  the  cause  of  the  difference  in  stability 
in  the  two  positions  ? 

d.  Experiment    with    either    block 
(both,  if  time  permits),  to  determine 
how  far  it  must  be  turned  on  any  of  its 
short  -edges   before   it   will   fall   over 
(Fig.  22).     For  this  purpose,  suspend 
the  plumb  line  from  the  nail  at  the 
center  of  gravity,  adjusting  the  length 
of  the  line  so  as  to  free  the  bob  from 
the  table   (or  stand  the  block  at  the 

edge  of  the. table,  so  that  the  plumb  line  hangs  free  in  front  of 
it),  and  observe  where  the  line  passes,  with  reference  to  the 
base  of  support,  at  the  instant  the  block  falls.  Write  your 
conclusion  in  the  form  of  a  general  statement  that  covers  all 
the  cases  that  you  have  tried. 

e.  Explain  this  behavior  of  bodies  in  overturning  them. 


UNBALANCED   FORCES:    DYNAMICS 

EXERCISE  20.     COMPARISON   OF  MASSES  BY 
INERTIA 

References.  —  Hoadley,  15,  26,  and  40-44 ;  Carhart  and  Chute, 
9,  13,  and  39-43 ;  Slate,  175-176 ;  Sanford,  pp.  26-28. 

To  measure  a  mass  by  the  effect  of  a  force  upon  it. 

Apparatus.  —  Torsion   apparatus  (Fig.  23),  with  weights  to 
fit  the  cups ;  clamp  to  fasten  it  to  the  table  or  other  support ; 
tumbler  or  beaker;  shot;  balance  and  weights. 
COLEMAN'S  PHY.  LAB.  MAN.  —  6 


66  MECHANICS   OF   SOLIDS 

[The  torsion  apparatus  consists  of  a  piece  of  spring  brass  or  steel  wire, 
about  No.  12  and  a  foot  or  more  in  length,  attached  rigidly  to  a  small 
block  at  the  upper  end  and  to  the  middle  of  a  light,  horizontal  bar  of  wood 
at  the  lower  end.  This  bar  is  8  or  10  in.  long,  and  carries  at  each  end  cups 
of  tin  or  brass  in  which  fit  snugly  equal  brass  weights.  Cylinders  of  lighter 
metal  than  brass  (as  zinc  or  iron)  are  to  be  preferred,  if  obtainable,  as 
the  distinction  between  equality  of  mass  and  of  volume  would  be  more 
strikingly  illustrated.  Weights  of  200  g.  are  perhaps  best  suited  to  the 
experiment,  but  either  100-g.  or  500-g.  weights  may  be  used.  For  500-g. 
weights  the  supporting  wire  should  be  No.  10  or  11.  The  wire  should  be 
of  such  size  and  length  that,  with  the  weights  used,  there  are  from  80  to 
120  single  vibrations  per  minute.] 

a.   Place  the  brass  weights  (hereafter  called  cylinders)  in 

the  cups,  and  set  the  bar  vibrat- 
ing in  a  horizontal  plane  by 
rotating  it  through  10°  or  15°. 
Count  the  number  of  single  vi- 
brations made  in  exactly  two 
minutes,  keeping  time  by  the 
second  hand  of  a  watch. 

b.  Set  the  bar  vibrating  through 
a  considerably  smaller  (or  larger) 
arc,  and  count  the  number  of 

vibrations  as  before.     What  effect  has  the  extent  of  swing,  or 
amplitude,  on  the  rate  of  swing  ? 

c.  Kemove  the  cylinders  from  the  cups  and  set  the  bar 
vibrating.  How  has  the  removal  of  the  cylinders  affected  the 
rate  ?  (The  rate  need  not  be  determined.)  The  vibration 
is  maintained  by  the  elasticity  of  the  supporting  wire;  and 
the  force  brought  into  action  by  twisting  it  is  the  same 
whether  the  cylinders  are  in  the  cups  or  not.  After  the 
cylinders  were  removed,  there  was  less  mass  (or  matter)  to 
be  moved  by  the  force;  hence  the  change  in  the  rate.  The 
weight  of  the  cylinders  is  not  what  affects  the  rate  of  vibra- 
tion. The  attraction  of  the  earth  plays  no  part  whatever  in 
the  experiment,  not  even  causing  friction,  as  is  generally  the 


COMPARISON   OF   MASSES   BY   INERTIA  67 

case  with  moving  bodies.  The  pull  of  the  earth  is  balanced 
by  the  tension  of  the  wire ;  the  motion  is  caused  by  the  torsion, 
which  is  entirely  independent  of  the  tension,  and  hence  of  the 
weight  also.  In  fact,  if  the  experiment  could  be  performed 
away  from  the  attraction  of  the  earth  or  of  any  heavenly 
body,  the  effect  of  the  cylinders  would  be  the  same  as  before, 
although  they  would  then  weigh  nothing. 

d.  From  the  above  determined  number  of  vibrations  in  2 
inin.  with  the  cylinders  in,  compute  the  number  of  vibrations 
in  1  min.,  and  also  in  30  sec.     These  numbers  are  wanted  for 
comparison  in  the  next  step. 

e.  Fill  the  cups  about  half  full  of  shot.     Set  the  apparatus 
vibrating  and  count  the  number  of  vibrations  in  30  sec.     If  the 
rate  is  faster  than  with  the  cylinders,  add  more  shot  to  each 
cup,  always  keeping  them  equally  full ;  if  the  rate  is  slower, 
remove   part  of   the    shot.     In   this  way,  by  repeated  trials, 
adjust  the  quantity  of  shot  till  the  rate  of  vibration  is  the 
same  as  with  the  cylinders.     When  a  difference  can  no  longer 
be  detected  in  30  sec.,  count  for  1  min.  and,  finally,  for  2  min. 
Time  is  saved  by  making  the  period  of  counting  short  at  first, 
and  greater  accuracy  is  secured  by  making  it  longer. 

/.  When  the  adjustment  called  for  has  been  made  as  closely 
as  possible,  unclamp  the  apparatus  and  empty  the  shot  into  the 
empty  tumbler  or  beaker,  keeping  a  hand  over  one  cup  to  avoid 
spilling  while  emptying  the  first  cup.  Weigh  the  shot  used. 

g.  The  experimental  errors  should  not  exceed  1%  or  2%. 
Within  this  limit,  how  do  the  masses  of  the  shot  used  and 
the  cylinders  compare  ?  Mass  is  really  measured  inertia  (Slate, 
224). 

Discussion.  —  This  experiment  teaches  that :  — 

(1)  The  inertia  of  a  body  is  not  to  be  confounded  with  its 
weight,  and  is  in  no  way  dependent  upon  it. 

(2)  When  equal  (unbalanced)  forces  produce  equal  effects 
upon  different  bodies,  these  bodies  have  equal  masses. 


MECHANICS    OF   SOLIDS 


(3)  When  equal  forces  act  upon  unequal  masses,  a  less  effect 
is  produced  upon  the  larger  mass.  That  is,  the  greater  the 
mass  of  a  body,  the  greater  its  inertia. 

Discuss  each  of  these  points,  showing  how  they  follow  from 
the  experiment. 

EXERCISE  21.     FALLING   BODIES:1    WHITING'S 
METHOD 

References.  —  Hoadley,  35-39,  78-79,  and  81 ;  Carhart  and 
Chute,  29-34  and  60-61. 

I.   To  find  the  acceleration  of  a  falling  body. 
Apparatus.  —  A   stick   suspended  to   swing  as  a  pendulum 
(Fig.  24);  meter  rod;   iron  or  lead  ball;   thread;   matches. 

[For  the  pendulum  use  a  stick  1.5  to  3  m.  long,  of  rectangular  cross 
section  about  2  by  4  cm.  Suspend  it  by  a  strip  of  canvas  or  leather,  with 

its  wider  side  turned  toward  the  suspended 

ball.  A  strip  of  carbon  paper  is  fastened 
at  top  and  bottom  of  the  pendulum,  with 
paper  beneath  to  receive  the  impression. 
The  ball  must  be  heavy  enough  to  hold  the 
pendulum  at  a  considerable  angle,  as  shown 
in  the  figure.  For  a  long  pendulum  a  lead 
ball  about  4  cm.  in  diameter  may  be  neces- 
sary. The  apparatus  must  be  so 'adjusted 
that  the  suspended  ball  just  touches  the 
pendulum  when  the  latter  hangs  vertically.] 

a.   Place   a   strip   of   white   paper 
under  the  carbon  paper  at  the  top  and 
bottom  of  the  pendulum.     Adjust  the 
ball  and  pendulum,  as  shown  in  the 
FIG.  24.  figure,  by  means  of  a  thread  passing 

1  Parts  II  and  III  may  be  omitted  if  a  pendulum  of  different  length 
is  not  provided.  Part  I  is  complete  in  itself.  If  Parts  II  and  III  are 
included,  the  exercise  is  complete  (from  a  different  point  of  view)  without 
Part  I.  d,  which  may  be  omitted. 


FALLING   BODIES:    WHITING'S   METHOD  69 

over  the  three  nails.  The  ball  should  hang  near  the  middle 
of  the  paper.  Without  disturbing  the  adjustment  of  the 
pendulum,  strike  the  ball  against  the  carbon  paper,  marking 
its  position  by  the  dot  thus  made  on  the  paper  beneath. 
When  the  ball  is  perfectly  at  rest  burn  the  thread  between 
the  upper  nails.  The  pendulum  should  strike  the  ball,  making 
a  dot  on  the  lower  paper.  Measure  the  distance  between  the 
upper  and  lower  dots. 

b.  Repeat  the  experiment ;   but   before  doing  so  mark  out 
the  dots  already  made  on  the  paper,  so  they  will  not  be  mis- 
taken for  the  new  ones.     It  is  better  to  replace  the  paper  by 
a  new  piece  after  a  few  trials.     If  the  second  result  does  not 
differ  by  more  than  1  cm.  from  the  first,  take  the  average  of 
the  two.     If  the  difference  is  greater  than  1  cm.,  make  fur- 
ther trials  till  you  get  three  or  more  results  agreeing  within 
1  or  2  cm.,  and  take  their  average.      Call  the  average  dis- 
tance S. 

c.  S  is  the  distance  the  ball  fell  while  the  pendulum  was 
swinging  to  a  vertical  position;  that  is  to  say,  while  it  was 
making  half  a  swing  or  vibration.     To  determine  this  time, 
set  the  pendulum  swinging,  and  count  the  number  of  swings 
it  makes  in  exactly  1  min.,  using  the  second  hand  of  a  watch. 
The  time  of  a  swing  does  not  change  as  the  arc  through  which 
the  pendulum  swings  grows  less ;  hence  from  the  number  of 
swings  in  60  sec.  you  can  determine  the  time  of  one  swing. 
Half  this  time  is  the  time  it  takes  the  ball  to  fall  S  cm. 
Compute  this  time  and  call  it  t. 

d.  Substitute  your  values  of  S  and  t  in  the  formula  for  fall- 
ing bodies  and  solve  it  for  g.    Compute  the  per  cent  of  error  of 
your  result. 

II.  To  find  the  relation  between  the  distance  and  the 
time  of  fall. 

Apparatus.  —  The  same  as  for  Part  T,  except  that  the  pendu- 
lum is  considerably  longer  (or  shorter). 


TO  MECHANICS   OF  SOLIDS 

a.  Repeat  all  the  work  of  Part  I  with  the  longer  (or  shorter) 
pendulum.     Let  the  distance  fallen  be  denoted  by  ^  and  the 
time  of  fall  by  tt. 

b.  Compute  the  ratios  S :  Sl9  t :  tly  (t :  ^)2  and  (t :  ^)3.    Assum- 
ing that  the  distance  fallen  by  a  freely  falling  body  starting 
from  a  state  of  rest  is  proportional  to  some  integral  power  of 
the  time  of  fall,  what  power  is  it,  as  indicated  by  these  ratios? 

c.  Compute  the  per  cent  of  difference  between  the  ratios  that 
should  be  equal. 

Mention  probable  sources  of  error  in  the  experiment. 

III.  To  compute  the  acceleration  of  a  falling  body 
from  the  data  of  Parts  I  and  II. 

a.  In  the  proportion  that  you  have  formed  from  the  quan- 
tities S,  $!,  t,  and  tl9  let  S  and  t  denote,  as  before,  the  distance 
and  time  respectively  for  Part  I,  and  let  Si  denote  the  distance 
fallen  when  ^  =  1  sec.     Substitute  these  values  in  the  propor- 
tion and  solve  it  for  S^ 

b.  Since  the  distance  fallen  in  one  second  from  a  state  of 
rest  is  one  half  the  acceleration  (g)  of  a  freely  falling  body,  we 

have  g  =  2Si  = cm. 

Compute  the  per  cent  of  error  of  your  experimental  value  of 
g,  assuming  the  true  value  to  be  980  cm. 

EXERCISE   22.     THE   SIMPLE   PENDULUM 

References.  —  Hoadley,  83-86 ;  Carhart  and  Chute,  68-73. 

Apparatus. — A  pendulum  stand  with  three  pendulums,  one 
with  wooden  and  two  with  iron  bobs  (Fig.  25) ;  watch  or  clock 
with  second  hand. 

[A  convenient  and  efficient  adjustable  suspension  is  shown  in  the  figure. 
A  slanting  notch  is  cut  to  receive  the  thread  at  an  angle  of  about  35°,  and 
a  cork  glued  above,  in  which  is  cut  a  slit  to  receive  the  thread.  The  fric- 
tion in  the  cork  holds  the  pendulum.] 


THE    SIMPLE   PENDULUM 


71 


I.  To    -find  whether  the  amplitude   of  a  pendulum 
affects  its  time  of  vibration,  or  period. 

a.  Adjust   the  pendulums  with  iron 
bobs  so  that,  when  started  together  with 
equal   amplitudes,  they  keep   together. 
It  is  better  to  have  the  pendulums  about 
as  long  as   the  apparatus  will  permit. 
The    adjustment    will     be     sufficiently 
accurate  if  one  has  not  visibly  gained 
on  the  other  in  half  a  minute.     After 
securing  this  adjustment,  start  the  pen- 
dulums together,  giving  one  an  amplitude 
of  4°  or  5°  and  the  other  an  amplitude 
about   one   third   as    great.      Let   them 
swing   together   for   half   a   minute   or 
more.     Does  one  gain  on  the  other? 

b.  Again    start    the    pendulums    to- 
gether, giving  one  an  amplitude  of  2° 
or  3°  and  the  other  an  amplitude  of  25° 
to   30°.      Observe  which   gains   on  the 
other.      Verify  the  result  by  a  second 
trial,  giving  the  larger  amplitude  to  the 
other  pendulum. 

How  does  a  large  amplitude  affect  the  period  of  a  pendulum? 

II.  To  find  whether  the  mass  and  material  of  a  pen- 
dulum affect  its  period. 

Adjust  the  pendulum  with  the  wooden  bob  to  the  same 
length  as  one  with  an  iron  bob.  The  pendulums  should  be 
long.  The  length  is  measured  from  the  point  of  support  to 
the  middle  of  the  bob.  Start  the  two  pendulums  together  with 
equal  amplitudes  (not  above  10°),  and  observe  whether  one 
gains  on  the  other  in  half  a  minute  or  more. 

How  do  you  find  the  period  of  a  pendulum  to  be  affected  by 
its  weight  or  the  material  of  which  it  is  made  ? 


FIG.  25. 


72 


MECHANICS   OF   SOLIDS 


III.  To  find  the  relation  between  the  period  of  a 
pendulum  and  its  length. 

Adjust  the  three  pendulums  to  lengths  having  the  ratio  of 
1,  ^,  and  %.  Lengths  of  36  in.,  9  in.,  and  4  in.  will  be  most  con- 
venient. Determine  the  number  of  vibrations  that  each  of  the 
pendulums  makes  in  exactly  60  sec.  Compute  for  each  of  the 
pendulums  the  value  of  the  ratio  VZ  :  t  (to  be  expressed  deci- 
mally). 

If  there  is  time,  test  the  ratio  of  the  periods  of  the  pendulums 
by  starting  together  the  36-in.  and  the  9-in.  pendulums,  and 
observing  how  many  swings  of  the  shorter  occur  during  one 
swing  of  the  longer.  Test  the  36-in.  and  the  4-in.  pendulums 
in  the  same  way. 

FORM  OF  BECORD  FOR  PART  III 


Length  =  L 

Whole  Time 

No.  of  Swings 

Time  of  1 
Vibration  =  t 

V^ 

\/2  :  t 

36  in. 

60  sec. 









9 

60 









4 

60 





— 



Discussion.  —  a.  The  ratios  VZ :  t  should  be  equal  within 
the  limits  of  experimental  errors  (1%  or  2%).  Compute  the 
per  cent  of  difference  between  the  greatest  arid  the  least  of 
them. 

b.  From  the  equality  of  these  ratios  determine  the  relation 
between  the  lengths  of  two  pendulums  (Lx  and  Z/2)  and  their 
periods  (^  and  £2). 

c.  How  should  the  pendulum  of  a  clock  be  adjusted  when  it 
is  losing  time  ?     When  it  is  gaining  time  ? 

d.  Why  does  a  pendulum  such  as  you  used  in  this  exercise 
come  to  rest  after  a  time  ? 

What  is  the  usual  shape  of  the  bobs  of  clock  pendulums  ? 
What  is  the  advantage  of  this  shape  ? 


THE    WHEEL   AND   AXLE 


73 


MACHINES:    WORK   AND    ENERGY 
EXERCISE   23.     THE   WHEEL  AND  AXLE 

References.  —  Hoadley,  92-106 ;  Carhart  and  Chute,  89-97. 
To  study  the  laws  of  the  wheel  and  axle. 

Apparatus.  —  Wheel  and  axle  mounted  in  any  convenient 
way,  and  with  cord  attached  to  the  axle  and  each  wheel ;  250-g. 
spring  balance  with  its  weight  recorded  on 
its  back ;  weights  of  about  1  kg.  and  2  kg. 
respectively  (2000-g.  balance  and  heavier 
weights  may  be  used) ;  meter  rod. 

a.  Weigh  the  small  weight  with  the  spring 
balance,  and  call  it  W.     Suspend  it  from 
the  axle,  and  tie  the  balance,  right  end  up, 
to  the  cord  on  the  smaller  wheel  (Fig.  26). 
Hold  the  balance  by  the  hook.     The  sus- 
taining force  (/)  includes  the  weight  of  the 
balance  (recorded  on  its  back)  in  addition 
to  the   scale   reading.      Observe   that  the 
latter  can  be  made  to  vary  appreciably  with- 
out causing  any  motion  of  the  apparatus. 
This  is  due  to  friction,  for  which  allowance 
must  be  made  in  order  to  find  what  the 
balancing  force  would  be  if  there  were  no 
friction.     This  is  accomplished  by  reading 

the  balance  while  slowly  and  steadily  raising  the  weight  and 
again  while  lowering  it.  Since  friction  always  opposes  motion, 
it  will  be  eliminated  by  taking  the  average  of  these  readings. 
To  this  average  add  the  weight  of  the  balance  for  /.  Record 
observations  as  indicated  below. 

b.  In  considering  the  condition  for  equilibrium  of  the  wheel 
and  axle,  it  is  to  be  regarded  as  a  modified  form  of  a  lever  of 
the  first  class,  as  shown  in  Figure  27,  where  A  denotes  the 


I 


FIG.  26. 


74  MECHANICS   OF   SOLIDS 

radius  of  the  axle  and  a  the  radius  of  the  wheel.  Measure 
(unless  given)  the  diameters  of  the  axle  and  the  wheel.  Write 
the  condition  for  equilibrium  and  see 
whether  your  observations  are  in  agree- 
ment with  it. 

c.  Eaise  the  weight  a  measured  distance 
(D),  not  less  than  30  cm.,  and  measure  the 
distance  (d)  through  which  the  force  acts 
(the  distance  between  the  positions  of  a 
corner  of  the  balance  before  and  after  the 
weight  was  raised).  Compute  the  work  done 
upon  the  weight  (  WD  )  and  the  work  done  by  the  applied  force 
(fd).  How  should  they  compare?  Find  the  per  cent  of  error. 
Record  measurements  and  computations  as  indicated. 

FORM  OF  RECORD 

a.  Weight  of  spring  balance  —  g.  (or  oz.) 
Weight  raised  (W)                                   — -  g.  (or  oz.) 
Reading  of  balance,  W  rising 

Reading  of  balance,  W  falling 
Sustaining  force  (/) 

b.  Radius  of  axle  (A)  =  —   —  cm.  (or  in.) 
Radius  of  wheel  (a) 

Moment  of  weight  (  WA )  = 

Moment  of  the  sustaining  force  (fa)  — 

Error 

Per  cent  of  error  -  % 

c.  Distance  weight  is  raised  (U)  —  cm.  (or  in.) 
Distance  through  which  /  acts  (d)  - 

Work  done  upon  the  weight  (Wdfy  =  -    —  kgm.  (or  ft.  Ib.) 

Work  done  by  /  (fd) 

Error 

Per  cent  of  error  = % 


THE   PULLEY  75 

Repeat  the  whole  experiment  with  the  larger  weight  and  the 
larger  wheel.  Make  a  section  drawing  and  in  it  indicate  the 
quantities  A,  a,  W,  /,  d,  and  D. 

Discussion.  —  a.  With  any  wheel  and  axle  the  applied  force 
will  support  a  weight  how  many  times  greater  than  itself? 
(See  paragraph  6.) 

b.  With  any  wheel  and  axle  the  applied   force  must  act 
,  through  a  distance  how  many  times  farther  than  the  weight  is 

lifted  ? 

c.  From  your  answers  to  the  preceding  questions  show  that 
the  work  done  upon  the  weight  (Wd)  must  always  be  equal 
to   the   work   done   by  the   applied   force   (fd),  disregarding 
friction. 

Taking  friction  into  account,  which  of  the  two  is  necessarily 
the  greater  ?  Why  ? 

EXERCISE  24.     THE   PULLEY 
References.  —  Hoadley,  107-110 ;  Carhart  and  Chute,  98-100. 

Apparatus.  —  A  single  and  two  double  or  triple  pulleys ; 
frame  with  screw  hooks  to  support  the  pulleys  (Fig.  28); 
meter  rod ;  a  weight  of  2  to  4  Ib.  and  one  of  6  to  16  Ib. ;  cord ; 
2000-g.  spring  balance. 

I.   To  study  the  laws  of  the  single  fixed  pulley. 

NOTE.  — Use  only  metric  or  only  English  units  throughout  the  exercise. 

a.  Suspend  the  pulley  and  pass  a  cord  over  it.  Attach  the 
smaller  weight  (  W  )  to  one  end  of  the  cord,  and  tie  the  balance, 
right  end  up,  to  the  other  end.  The  force  (/)  necessary  to 
balance  the  weight  equals  the  reading  of  the  balance  plus  its 
weight  (recorded  on  the  back).  Observe  that  the  scale  reading 
can  be  made  to  vary  appreciably  without  moving  the  weight. 
This  is  due  to  friction,  for  which  allowance  must  be  made  in 


76 


MECHANICS   OF   SOLIDS 


order  to  find  what  the  sustaining  force  would  be  if  there  were 
no  friction.  To  eliminate  the  effect  of  friction,  read  the  balance 
while  slowly  and  steadily  raising  the  weight  and  again  while 
lowering  it,  and  take  the  average.  To  this  average  add  the 
weight  of  the  balance.  Kecord  W.  In  the  form  of  record  n 


S 


-A\ 


FIG.  28. 

denotes  the  number  of  parts  of  the  cord  supporting  the  weight. 
(In  this  case  n  =  1.) 

b.  Starting  with  the  balance  near  the  pulley,  pull  it  down 
through  a  measured  distance  of  50  fern,  to  80  cm.,  and  measure 
the  distance  the  weight  rises.  The  former  is  recorded  as  the 
distance  through  which  the  force  acts  (d),  and  the  latter  as  the 
distance  through  which  the  weight  moves  (Z>). 


THE   PULLEY  77 

Compute  the  work  done  by  the  force  (fd)  and  the  work  done 
upon  the  weight  (WD),  using  the  kilogram-meter  (or  foot-pound) 
as  the  unit.  How  do  they  compare  ? 

What  advantage  is  there  in  using  a  single  fixed  pulley  ? 

II.  To  study  the  laws  of  the  single  movable  pulley. 

a.  Weigh  the  single  pulley  and  adjust  it  as 
shown  in  Fig.  29,  using  the  larger  weight  if  not 
greater  than  4  kg.     With  this  adjustment,  W  in- 
cludes the  weight  of  the  pulley.     Does  /  include 
the  weight  of  the  balance  ?     In  determining  / 
eliminate  friction  as  before.     Record  /,  W,  and 
n  (n  =  2). 

What  relation  is  there  between  the  mechanical 
advantage  ( W :  /)  and  n  ?  Account  for  this 
relation,  bearing  in  mind  that  the  tension  is  the 
same  in  all  parts  of  the  cord. 

b.  Starting  with  the   weight   low,   raise  the 
balance  through  a  measured  distance  not  less 
than  50  cm.     How  far  is  the  weight  lifted  ? 

Discover  the  connection  between  the  ratio  of 
these  distances  and  the  number  of  parts  of -the 
cord  supporting  the  weight. 

What  is  gained  by  using  a  movable  pulley  ?  G' 

Make  a  section  drawing  and  in  it  indicate  /,  TF,  d,  and  D. 

III.  To    study    combinations    of  fixed    and    movable 
pulleys. 

a.  With  the  two  double  or  triple  pulleys  arrange  any  com- 
bination that  you  may  choose.      Use  the  heavier  weight  or, 
if  convenient,  both.     Weigh  the  movable  pulley  and  remember 
to  include  it  as  part  of  W.      Follow  directions  and  answer 
questions  of  Part  II.  a. 

b.  Pull  the  balance  down  through  a  measured  distance  of 
50  cm.  or  more,  and  measure  the  distance  through  which  the 


78 


MECHANICS   OF   SOLIDS 


weight  is  lifted.     Answer  the  questions  of  Part  II,  b.     Make  a 
drawing  of  the  arrangement  of  pulleys. 

c.  If  time  remains,  arrange  other  combinations  of  pulleys, 
and  record  the  observations,  together  with  drawings  of  the 
combinations. 

FORM  OF  RECORD 


SCALE  READING 

/ 

W 

n 

Wif 

ERRORS 
n-W  :f 

%OF 

ERROK 

t'p 

Down 

I 

—  g. 

g- 

g. 

g- 







% 

II 

— 















III 

— 













d 

'D 

WORK  DONE                   j   ERRQR 

%OF 

By/  (-/<*) 

Upon  W(=\VD)  Jd    WI) 

ERROR 

I 

cm. 

cm. 

kgrn. 

o/ 

10 

II 







I  



III 











Discussion.  —  a.  Draw  a  figure  of  a  single  fixed  pulley  with 
a  force  /  supporting  a  weight  W.  Draw  the  horizontal  diameter 
of  the  pulley ;  and  regard  this  as  a  lever  and  the  axis  of  the 
pulley  as  its  fulcrum.  Express  the  condition  for  equilibrium 
and  show  why  /  must  be  equal  to  W. 

b.  With  any  combination  of  pulleys  a  given  force  will  sup- 
port a  weight  how  many  times  greater  than  itself  ?     Why  ? 
Word  your  answer  so  as  to  apply  to  all  cases. 

c.  With  any  combination  of  pulleys  the  applied  force  must 
act  through  a  distance  how  many  times  farther  than  the  weight 
is  raised  ?     Why  ? 

d.  From  your  answers  to  b  and  c  show  that,  neglecting  friction, 
there  is  neither  saving  nor  loss  of  work  by  the  use  of  pulleys. 

What  is  the  advantage  gained  by  their  use  ? 


THE    INCLINED   PLANE 


79 


EXERCISE   25.     THE   INCLINED   PLANE 
References.  —  Hoadley,  111 ;  Carhart  and  Chute,  101-102. 

Apparatus.  —  An  inclined  plane  (Fig.  30) ;  roller  or  car ; 
250-g.  spring  balance. 

I.  To  find  the  ratio  of  the  weight  to  the  force  neces- 
sary to  sustain  it  when  the  latter  is  applied  parallel 
to  the  plane. 

a.  Set  up  the  plane  at  an  angle  of  about  20°.  It  is  not 
necessary  to  have  the  angle  exact  or  to  measure  it.  Measure 
the  height  (H)  and  the  length  (L)  of  the  plane.  Since  we 
need  only  the  ratio  of  these  quantities,  we  may  take  for  meas- 


urement the  right  triangle  formed  by  the  under  edge  of  the 
plane,  the  upper  edge  of  the  base,  and  the  straight  vertical  edge 
of  the  support  of  the  plane.  With  the  balance  determine  the 
force  (/),  parallel  to  the  plane,  necessary  to  hold  the  roller  (or 
car)  in  equilibrium  on  the  plane.  This  may  be  taken  as  the 


80 


MECHANICS   OF   SOLIDS 


average  of  the  readings  when  the  roller  is  moving  slowly  up 
the  plane  and  slowly  down  it.  Friction  is  thus  eliminated. 
Weigh  the  roller  and  call  its  weight  W. 

b.  Repeat  the  set  of  measurements  with  the  plane  at  a  con- 
siderably greater  angle.  (The  angle  must  not  be  so  large  that 
/exceeds  the  maximum  reading  of  the  balance.) 

II.  To  find  the  ratio  of  the  weight  to  the  force  neces- 
sary to  sustain  it  when  the  latter  is  applied  parallel 
to  tlw  base. 

a.  Take  a  set  of  readings  with  the  applied  force  parallel  to 
the  base,  and  the  plane  at  the  same  angle  as  for  I  6?  unless  the 
force  needed  is  greater  than  the  balance  will  register,  in  which 
case  lower  the  plane.      Instead  of  L  record  the  length  of  the 
base  (B). 

b.  For  a  given  angle  of  the  plane,  is  the  applied  force  greater 
when  applied  parallel  to  the  plane  or  parallel  to  the  base  ? 

FORM  OF  RECORD 


SCALE  READING 

f 

W 

ff 

Up 

Down 

I  a 

—  —  —  £f. 

g. 

g- 

g- 

L  =  cm. 

cm. 

b 









L  =  cm. 



II  a 







B=  cm. 



W:f 

ERROR 

%  OF  ERROR 

I  a 



L:  H=  

W: 

f-L:  PI=  

o/o 

r 

L  -  H  — 

W  - 

/     £  '  H  — 

II  a 



B:  H=  

W: 

f             J)   .     TT  



THE   INCLINED   PLANE  81 

Discussion.  —  a.  Make  a  section  drawing  of  the  plane,  and  in 
it  indicate  the  quantities  /,  W,  B,  L,  and  H.  Resolve  W  into 
components  respectively  perpendicular  and  parallel  to  the  in- 
clined plane.  Write  the  relation  that  holds  among  the  quan- 
tities f9  W)  L,  and  H.  Test  your  results  by  this  relation,  as 
indicated  in  the  form  of  record. 

b.  Make  a  similar  drawing  in  which  W  is  resolved  into  com- 
ponents respectively  perpendicular  to  the  inclined  plane  and 
parallel  to  the  base ;  and  write  the  relation  that  holds  for  f9 
W,  B,  and  H. 

c.  A  plane  is  inclined  at  an  angle  of  30°.    What  is  the  value 
of  L  :  H?     What  force  parallel  to  the  plane  would  be  neces- 
sary to  support  a  200-lb.  barrel  on  it  ? 

d.  How  much  work  would  be  done  in  rolling  the  barrel  10  ft. 
up  the  plane  ?      Through  what  vertical  distance   would  the 
barrel  be  raised  ? 

e.  How  much  work  would  have  been  done  if  the  barrel  had 
been  lifted  vertically  that  distance  without  the  aid  of  any 
machine  ? 

/  What  advantage  is  derived  from  the  use  of  the  plane  ? 


COLEMAN'S  PHY.  LAB.  MAN.  —  6 


V.     HEAT 
EXERCISE  26.      EXPANSION   BY   HEAT 

References.— Hoadley,  235-238 ;  Carhart  and  Chute,  305-307. 

I.  To  observe  and  compare  the  effect  of  heat  on  brass 
and  iron. 

Apparatus.  —  Ball  and  ring  of  brass;  Bunsen  burner;  jar  of 
water ;  compound  bar  of  iron  and  brass  riveted  together. 

a.  To  heat  either  the  ball  or  ring,  hold  it  in  the  Bunsen  flame 
about  a  minute ;  to  cool  them,  thrust  them  into  the  jar  of  water. 
Do  not  lay  the  hot'  ball  or  ring  on  the  table.  Find  by  trial 
whether  the  ball  will  pass  through  the  ring,  —  (1)  when  both 
are  cold ;  (2)  when  the  ball  is  hot  and  the  ring  cold ;  (3)  when 
both  are  hot. 

State  and  account  for  the  result  in  each  case. 

6.  Heat  both  strips  of  the  compound  bar  as  nearly  equally 
as  possible.  Hold  them  in  the  flame  so  that  they  are  side  by 
side,  not  one  above  the  other.  What  does  the  bending  of  the 
bars  indicate  ? 

II.  To  observe  the  effect  of  heat  on  the   volume  of 
water. 

Apparatus A  small  flask  or  bottle  of  water,  fitted  with 

stopper  and  small  glass  tube,  with  small  rubber  band  for  an 
index  (narrow  section  of  small  rubber  tubing)  ;  vessel  for  heat- 
ing water ;  Bunsen  burner. 

Heat  the  vessel  of  water  about  as  hot  as  you  can  bear  with 
your  hand.  See  that  the  water  stands  a  few  centimeters  high 
in  the  glass  tube  inserted  in  the  flask  (Fig.  31).  Its  height 
can  be  increased  by  pushing  the  stopper  in  farther.  Mark  the 

«»  82 


CONDUCTION   OE   HEAT 


83 


FlG  3 


height  of  the  water  by  the  rubber  band  on  the  tube. 
Lower  the  flask  of  water  into  the  hot  water,  keep- 
ing a  sharp  watch  for  the  first  motion  of  the  water 
in  the  tube.  State  and  account  for  the  observed 
changes  of  level. 

III.    To  observe  the  effect  of  gain  and  loss 
of  heat  on  the  volume  of  air. 

Apparatus.  —  Small  flask  with  stopper  and  glass 
tube  inserted  ;  tumbler  of  water. 

a.  With  the  flask  inverted,  thrust  the  end  of  the 
glass  tube  into  the  tumbler  of  water  ;  and  inclose 
the  flask  in  the  hands  so  as  to  heat  it  and  the  air 
inside.     Observe  whether  bubbles  rise  in  the  tum- 
bler ;  and  if  they  do,  account  for  them. 

b.  With  the  tube  still  in  the  water,  remove  the 

hands  from  the  flask  and  observe  any  movement  of   air  or 
water  in  the  tube.     Explain. 

c.  Which  was  raised  to  the  higher  temperature,  the  flask  of 
water  or  the  flask  of  air  ? 

From  your  observations  which  do  you  think  would  expand 
more  for  equal  changes  of  temperature,  water  or  air  ? 

EXERCISE   27.     CONDUCTION  OF  HEAT 

References.  —  Hoadley,  253-254;  Carhart  and  Chute,  343- 
347  ;  Sanford,  p.  170  ;  Jones,  2  ;  Madan,  pp.  2-5  and  252-253. 

I.  To  determine  the  order  in  which  glass  and  different 
metals  stand  as  conductors  of  Iwcut. 

Apparatus.  —  Rods  of  brassy  iron,  glass,  and  copper  (use  wires 
of  the  different  metals  of  about  No.  9  to  12)  ;  Bunsen  burner  ; 
vessel  of  water  ;  test  tube. 

a.  Test  the  relative  conductivity  of  the  rods,  two  at  a  time, 
holding  one  in  each  hand,  with  an  end  of  each  in  the  Bunsen 
flame  about  6  cm.  above  the  burner  ;  the  two  being  as  nearly  as 


84  HEAT 

possible  equally  heated.  At  first  hold  the  rods  near  the  heated 
end ;  and,  as  they  become  uncomfortably  hot,  hold  them  farther 
from  this  end,  observing  in  which  the  heat  travels  faster.  The 
method  may  be  varied  by  holding  the  rods  at  the  same  distance 
from  the  heated  end,  and  observing  in  which  the  heat  first 
reaches  the  hand. 

Before  trying  the  same  rod  a  second  time,  it  may  be  cooled 
in  the  vessel  of  water ;  except  the  glass  rod,  which  must  be 
allowed  to  cool  of  itself,  for  it  will  break  if  thrust  into  the 
water  hot.  Continue  experimenting  till  you  can  arrange  the 
rods  in  the  order  of  their  conductivity.1  Write  the  names  in 
order,  from  the  best  to  the  poorest  conductor. 

b.  Fill  the  test  tube  with  water  within  an  inch  of  the  top. 
Hold  it  at  the  bottom  with  the  fingers,  tipping  it  slightly,  and 
apply  the  Bunsen  flame  a  little  below  the  top  of  the  water,  till 
it  boils  at  the  top  for  about  a  minute. 

What  do  you  observe  concerning  the  temperature  of  the 
water  at  the  bottom  of  the  tube  ? 

What  do  you  conclude  concerning  the  conductivity  of  water? 

II.  To  observe  whether  sensations  of  heat  and  cold  are 
affected  by  the-  conductivity  of  the  substance  touched. 

Apparatus.  —  Some  or  all  of  the  following  substances,  heated 
to  the  same  temperature  in  an  air  bath :  brass,  iron,  glass,  cop- 
per, wool,  asbestos,  stone,  wood.  The  same  substances,  cooled 
to  equal  temperatures  in  an  ice  box. 

a.  Eemove  from  the  air  bath  and  cautiously  feel  the  differ- 
ent substances,  two  at  a  time,  one  in  each  hand.  Arrange 
them  as  nearly  as  you  can  in  order,  beginning  with  the  one 
that  feels  the  hottest. 

1  The  rise  of  temperature  along  the  rods  depends  upon  another  property 
of  the  substances  (not  yet  studied)  besides  their  conductivity.  But  this 
property  (specific  heat)  would  not  affect  the  order  of  conductivity  of  the 
materials  used  in  this  experiment. 


CONVECTION   OF   HEAT  85 

These  substances  were  really  equally  hot,  all  being  at  the 
temperature  of  the  hot  air  of  the  bath,  to  which  they  had  been 
exposed  for  a  considerable  time.  Account  for  their  seeming 
difference  of  temperature. 

b.  Feel  the  same  substances  cooled  in  the  ice  box,  and  arrange 
in  order,  beginning  with  the  one  that  feels  the  coldest.  In 
doubtful  cases  press  the  substances,  two  at  a  time,  against  the 
forehead,  which  is  more  sensitive  than  the  hands. 

Were  their  temperatures  really  different  ?     Explain. 

Discussion.  —  a.  Why  is  woolen  clothing  warmer  than  cotton 
or  linen  ? 

b.  Why  is  a  carpet  more  comfortable  to  the  bare  feet  in 
cold  weather  than  the  floor  ? 

c.  Why  is  asbestos  used  as  wrapping  for  steam  pipes  and 
boilers  ? 

d.  An  overcoat  is  said  to  "  keep  out  the  cold."     What  is  it 
that  it  really  does  ? 

EXERCISE  28.  CONVECTION  OF  HEAT 

References.  —  Hoadley,  255-257;  Carhart  and  Chute,  348- 
349. 

I.   To  study  convection  currents  in  water. 

Apparatus.  —  Test  tube;  Bunsen  burner;  beaker;  sawdust; 
iron  stand ;  wire  gauze  ;  mop  cloth. 

a.  Put  a  very  little  sawdust  in  the  beaker,  and  fill  it  nearly 
full  of  water.  Wipe  the  outside  of  the  beaker  dry,  place  it 
on  the  wire  gauze  on  the  ring  stand,  and  apply  heat  with  the 
Bunsen  flame.  The  gauze  should  be  about  10  cm.  above  the 
burner,  and  the  flame  turned  down  so  that  it  will  not  burn 
above  the  gauze.  Observe  carefully  the  motion  of  the  particles 
of  sawdust  while  the  water  is  heating.  What  does  this  motion 
indicate  ? 


HEAT 


Give  a  full  account  of  what  is  observed,  including  definite 
reasons  for  any  motion  of  the  liquid  that  you  may  infer  from 
the  behavior  of  the  sawdust.  (Consider  the  results  of  Part  II 
of  Exercise  26.) 

6.  Fill  the  test  tube  nearly  full  of  water,  hold  it  in  the  hand 
just  below  the  surface  of  the  water,  and  apply  the  flame  near 
the  bottom.  Note  the  rapidity  of  the  rise  of  temperature 
where  held.  Compare  with  the  experiment  in  conduction  in 
which  the  heat  was  applied  near  the  top  of  the  tube  and  was 
held  near  the  bottom. 

Account  for  the  difference  in  the  results  of  the  two  experi- 
ments. 

II.   To  study  convection  currents  of  air. 

Apparatus.  —  Two  student  lamp  chimneys ;  a  chalk  or  paste- 
board box,  with  lidr  that  fits  closely,  and  with  a  hole  about  an 
inch  in  diameter  in  the  lid  near  one  end,  and  a  group  of  small 
holes  that  can  be  covered  by  the  chimney  near  the  other  end 
(Fig.  32) ;  candle ;  touch-paper ;  jar  of 
water ;  cloth  and  stick. 

[Touch-paper  is  made  by  soaking  filter 
paper  in  a  strong  solution  of  saltpeter  and 
drying.] 

a.  Light  the  candle  and  place  it  on 
the  box  over  the  group  of  small  holes, 
and  place  a  student  lamp  chimney 
over  it,  being  careful  to  cover  all  the 
holes  with  the  chimney.  Any  wax  or 
tallo^  that  keeps  the  chimney  from 

fitting  tightly  must  be  scraped  off.  Place  the  other  chimney 
over  the  large  hole,  and  study  the  air  currents  by  holding 
burning  touch-paper  over  the  tops  of  the  chimneys.  When  the 
paper  has  burned  nearly  to  the  fingers,  throw  it  into  the  jar  of 
water.  Describe  and  account  for  the  currents  of  air  discovered. 
b.  Observe  the  effect  upon  the  candle  when  the  chimney  is 


RADIANT   ENERGY  87 

removed  from  the  large  hole  and  the  hole  tightly  covered  with 
the  hand.  Describe  the  result  and  explain  it.  If  the  burning 
paper  has  soiled  the  chimneys,  clean  them  with  the  cloth  and 
stick.  Leave  the  apparatus  and  the  table  clean  and  in  order. 

36.  Radiant  Energy.  —  Tt  should  be  understood  that  so-called 
"  radiant  heat "  is  not  heat  at  all,  but  a  wholly  different  form 
of  energy  which  is  readily  transformable  into  heat  by  absorp- 
tion, —  a  process  analogous  to  the  transformation  of  the  kinetic 
energy  of  a  flying  bullet  into  heat  when  it  strikes  a  steel  target. 
Hence,  of  course,  the  "  radiation  of  heat "  is  not  a  process  of 
heat  transmission,  but  is  the  transmission  of  this  other  form 
of  energy.  Its  correct  name  is  radiant  energy,  and  the  process 
of  transmission  is  radiation  (not  radiation  of  heat). 

But  light  is  also  radiant  energy,  and  its  transmission  is  called 
radiation ;'  and  it  is  desirable  to  be  able  to  distinguish  without 
too  many  words  between  light  and  so-called  "  radiant  heat." 
Since  the  latter  does  not  affect  the  eye,  it  is  appropriately 
called  invisible  radiation;  and  is  thus  distinguished  from  light, 
which  is  visible  radiation. 

In  general,  the  energy  of  invisible  radiation  is  greater  than 
that  of  light,  and  hence  has  greater  heating  power  when  ab- 
sorbed ;  but  the  energy  of  light  is  also  transformed  into  heat 
by  absorption. 

EXERCISE   29.     RADIANT   ENERGY 

References.— Hoadley,  258-263;  Carhart  and  Chute,  350- 
354 ;  Slate,  154-155 ;  Sanf ord,  pp.  17  -177 ;  Jones,  86,  92. 

Apparatus.  —  Radiometer;  Bunsen  burner ;  pasteboard  screen 
about  8  in.  wide  and  10  in.  high,  mounted  on  block;  two 
mounted  tin  screens,  one  bright,  the  other  painted  black  or 
coated  with  soot  on  one  side ;  three  flat  bottles  of  clear  glass, 
one  empty,  one  filled  with  water,  and  one  with  a  solution  ©f 
iodine  in  carbon  bisulphide. 


88  HEAT 

I.  To  study  radiation  and  to  distinguish  it  from  con- 
duction and  convection. 

a.  Hold  the  hands  beside  the  Buusen  flame  at  different  dis- 
tances and  above  it.     Note  the  intensity  of  the  sensation  of 
heat  in  the  different  positions ;  also  note  whether  you  can  feel 
convection  currents  in  any  position.     In  what  position  does 
the  hand  receive  heat  by  convection  ? 

b.  Hold  the  hand  beside  the  flame  and  within  a  few  inches 
of  it,  and  note  the  intensity  of  the  sensation.      Insert  the 
pasteboard  screen  between  your  hand,  still  in  this  position, 
and  the  flame.     How  does  this  affect  the  sensation  of  heat  ? 

If  the  heat  came  to  the  hand  by  conduction  or  convection, 
could  it  get  round  the  screen  ? 

State  all  the  facts  pointing  to  the  conclusion  that  the  hand 
is  not  heated  by  conduction  or  convection  when  it  is  beside  the 
flame. 

Does  the  radiation  reach  the  hand  by  a  straight  or  a  bent  path  ? 

c.  Place  the  radiometer  at  different  distances  from  the  flame, 
and  observe  the  effect  of  distance  upon  the  rate  of  rotation  of 
the  vanes.     Eadiation,  both  visible  and  invisible,  falling  upon 
the  radiometer,  will  cause  the  vanes  to  rotate ;  and  the  rate  of 
rotation  is  an  indication  of  the  energy  of  the  radiation.     How 
this  effect  is  produced  need  not  concern  you  here. 

d.  Place  the  radiometer  25  or  30  cm.  from  the  flame  and 
slowly  insert  the  pasteboard  screen  between  them.     What  is 
the  position  of  the  screen  when  the  slower  rotation  of  the 
vanes  indicates  that  the  radiation  has  been  cut  off  from  the 
radiometer  ? 

What  evidence  does  this  afford  on  the  question  how  radia- 
tion travels  ? 

II.  To  test  the  power  of  different  substances  to  absorb 
and  transmit  visible  and  invisible  radiation. 

a.  Place  the  tin  screens  on  opposite  sides  of  the  flame,  about 
10  cm.  from  it,  with  the  black  side  of  the  one  toward  the  flame. 


RADIANT   ENERGY  89 

After  a  minute  or  two,  note  the  temperatures  of  the  screens  by 
placing  a  hand  flat  against  each  on  the  side  turned  from  the 
flame.  State  and  account  for  what  is  observed. 

b.  Eemove  the  bright  screen  and  hold  the  hand  in  its  place 
at  the  same  distance  from  the  flame  as  the  other  hand,  which 
is  still  held  against  the  back  of  the  black   screen.     Which 
hand  becomes  warmer  ?     Explain. 

Note  and  account  for  the  difference  of  temperature  of  the 
palm  and  back  of  the  hand  that  receives  the  direct  radiation. 

Do  you  think  that  the  palm  or  back  of  the  hand  or  the  black 
screen  is  more  nearly  at  the  temperature  of  the  air  at  that  dis- 
tance from  the  flame  ?  Give  reasons  for  your  opinion. 

c.  Hold  the  empty  flask  between  the  flame  and  the  radi- 
ometer, and  bring  the  latter  up  till  the  vanes  make  about  one 
rotation  per  second.     Now  remove  the  flask  and  note  the  effect 
on  the  radiometer.     What  do  you  infer  in  regard  to  the  be- 
havior of  clear  glass  toward  radiation  ? 

d.  Without  moving  the  radiometer  or  the  flame,  hold  the 
flask  of  water  between  them.     Compare  the  result  with  that 
obtained  with  the  empty  flask.      A  more  definite  comparison 
may  be  made  by  observing  the  time  of,  say,  ten  rotations; 
but  where  there  are  easily  observable  differences  this  is  not 
necessary. 

e.  Substitute  the  flask  containing  the  solution  of  iodine  in 
carbon  disulphide.     Compare  the  result  with  the  preceding. 

Does  the  solution  transmit  visible  radiation  ?  (Can  you  see 
through  it  ?) 

What  evidence  is  there  that  it  transmits  invisible  radiation  ? 

Is  it  a  better  or  a  poorer  transmitter  of  light  than  water  ? 
Is  it  a  better  or  poorer  transmitter  of  invisible  radiation  ? 

Substances  that  transmit  light  readily  are  called  transparent; 
those  that  do  not  transmit  light  are  called  opaque.  Substances 
that  readily  transmit  invisible  radiation  are  called  diaiher- 
manous;  those  that  absorb  instead  of  transmitting  it  are 
called  athermanous. 


90  HEAT 

III.   To  find  in  what  direction  radiation  is  reflected 
by  a  smooth  surface. 

a.  Adjust  the  flaine  (F),  the  pasteboard  screen  (AB),  and 
the  bright  tin  screen  (CD)  in  the  relative  positions  shown  in 

Fig.  33.     AB  should  be  within  10  or  12  cin. 

C  D 

of  the  flame  and  within  2  cm.  of  CD.  Com- 
pare the  rates  of  rotation  of  the  radiometer 
at  a,  b,  and  c.  Observe  that  the  flame  and 
the  radiometer,  when  the  latter  is  at  a,  are 
symmetrically  situated  with  respect  to  the 
B  screens.  With  the  radiometer  at  a,  try  the 

effect  of  turning  CD  at  different  angles, 
always  keeping  it  near  the  edge  of  AB.  On  what  part  of  CD 
does  the  radiation  fall  that  is  reflected  into  the  space  CAB  ? 

What  does  this  experiment  show  concerning  the  law  of  reflec- 
tion of  radiation  ? 

b.  Eeplace  CD  by  the  black  screen,  using  the  black  surface. 
Place  the  radiometer  at  a.     State  and  account  for  the  result. 

c.  Compare  the  reflecting  powers  of   the  two  screens  and 
their  powers  of  absorption  as  determined  in  II  a. 

What  relation  is  shown  between  absorbing  and  reflecting 
powers  ?     Account  for  this  relation. 


EXERCISE  30.      COEFFICIENT    OF    LINEAR 
EXPANSION 

References.  — Hoadley,  264-265;  Carhart  and  Chute,  319. 

To  find  the  expansion  of  1  cm.  of  a  brass  rod  for  T 
rise  of  temperature. 

Apparatus. — Linear  expansion  apparatus  (Fig.  34);  appa- 
ratus for  generating  steam  ;  access  to  a  thermometer;  tumbler; 
meter  rod ;  Bunsen  burner. 

[For  the  steam  generator  use  a  copper  boiler  on  tripod,  with  tight  top, 
or  flask  with  stopper  and  delivery  tube,  supported  on  a  ring  stand.] 


COEFFICIENT   OF   LINEAR   EXPANSION 


91 


a.  Fill  the  steam  generator  from  one-third  to  one-half  full 
of  water,  and  with  the  top  off  (or  the  delivery  tube  discon- 
nected at  the  generator)  begin  heating  it.  While  the  water  is 
heating,  measure  the  length  of  the  brass  rod  without  removing 
it  from  the  steam  jacket ;  then  adjust  it  so  that  one  end  rests 
against  the  fixed  support  and  the  other  against  the  lever. 
Turn  it  so  that  the  escape  tube  will  be  directed  downward. 
Set  the  tumbler  under  this  tube  to  catch  the  escaping  steam 
and  hot  water. 

6.  Read  to  .1  mm.  the  position  of  the  top  of  the  long  lever 
arm  on  the  vertical  scale  near  its  end.  After  taking  this  read- 
ing be  careful  not  to  disturb  the  apparatus,  as  any  change  in 
the  relative  position  of  the  parts  (for  example,  a  slight  rotation 


of  the  steam  jacket)  may  cause  an  appreciable  change  in  the 
reading  just  taken. 

c.  The  temperature  of  the  rod  is  the  same  as  that  of  the 
room.     Find  it  by  the  laboratory  thermometer. 

d.  Put  the  top  on  the   steam    generator  and  connect  the 
delivery  tube.     While  the  rod  is  being  heated  by  the  steam, 
observe  the  motion  of  the  long  lever  arm.     After  the  steam 
has  been  escaping  freely  from  the  escape  tube  for  two  or  three 
minutes  and  no  further  motion  of  the  rod  can  be  detected,  read 
the  position  of  the  long  lever  arm.     The  temperature  of  the 
rod  is  the  same  as  the  temperature  of  the  steam,  which  may 
be  assumed  to  be  100°. 

e.  Measure  the  arms  of  the  lever.     These  are  the  distances 
from  the  fulcrum  (the  center  of  the  screw)  to  the  scale  and 


92  HEAT 

from  the  fulcrum  to  the  point  of  contact  with  the  rod  respec- 
tively. When  you  have  finished,  tilt  the  apparatus  so  that 
the  water  condensed  in  the  steam  jacket  will  run  out. 

In  finding  the  whole  expansion  of  the  rod,  called  for  in  the 
computations,  make  use  of  the  fact  that  the  ends  of  the  lever 
arms  move  through  distances  that  are  proportional  to  the 
lengths  of  the  arms. 

OBSERVATIONS 

a.  Length  of  brass  rod  = cm. 

b.  First  position  of  long  lever  arm  = cm. 

c.  First  temperature  of  the  rod        =—   -  °  C. 

d.  Final  temperature  of  the  rod       = °  C. 

Final  position  of  long  lever  arm  = cm. 

e.  Length  of  long  lever  arm  = cm. 

Length  'of  short  lever  arm  = cm. 

COMPUTATIONS 

Change  of  temperature  of  rod  = °  C. 

Expansion  of  the  rod  for  this  change  of  temp.  = cm. 

Expansion  of  the  rod  for  1°  change  of  temp.     = cm. 

Expansion  of  1  cm.  of  rod  for  1°  change  of  temp.  = cm. 

The  last  quantity  is  the  coefficient  of  linear  expansion  of 
brass,  the  correct  value  of  which  is  .0000188.  Compute  the 
per  cent  of  error  of  your  result. 

ALTERNATIVE  DIRECTIONS 

If  the  apparatus  is  provided  with  a  micrometer  screw  instead 
of  a  lever,  the  following  modifications  of  the  directions  will 
apply  :- 

b.  If  you  do  not  know  how  to  read  the  micrometer  screw, 
turn  it  back  and  forth  and  study  its  action.  Note  the  fixed 
millimeter  scale  and  the  circular  scale  on  the  head ;  also  that 
when  the  head  is  turned  once  round  it  advances  1  mm.  along 


COEFFICIENT   OF  EXPANSION   OF   LIQUIDS  93 

the  fixed  scale.  How  many  divisions  are  there  on  the  circular 
scale  ?  The  value  of  one  division  on  the  circular  scale  is  the 
distance  the  screw  advances  when  the  head  is  turned  through 
one  division.  What  is  this  value  ?  Ask  for  assistance  if  neces- 
sary. The  answers  to  these  questions  need  not  be  recorded. 

c.  Turn  the  micrometer  screw  till  it  just  touches  the  rod, 
and  take  its  reading.     After  taking  the  reading,  turn  the  screw 
back  2  or  3  mm.  to  make  room  for  the  expansion  of  the  rod  when 
heated.     If  this  precaution  is  not  taken,  the  expanding  rod  will 
strain  and  damage  the  apparatus.    Eead  the  additional  precau- 
tion in  paragraph  b  above. 

d.  In  this  paragraph  substitute  for  the  reading  of  the  lever 
a  second  reading  of  the  screw,  after  it  has  been  turned  up  to 
touch  the  rod.     Omit  paragraph  e. 


EXEKCISE  31.     COEFFICIENT  OF  EXPANSION  OF 
LIQUIDS 

References.  —  Hoadley,  266-267;  Carhart  and  Chute,  317, 
319. 

Apparatus.  —  Flask  fitted  with  stopper  and  delivery  tube  and 
supported  on  stand  (Fig.  35) ;  Bunsen  burner ;  three  hydrom- 
eter jars;  stirrers;  thermometers,  each  with  a  tube  attached 
containing  the  liquids  for  the  exercise  (Fig.  36) ;  ice. 

[For  the  tube  to  contain  the  liquids  take  a  piece  of  quarter  inch  glass 
tubing  a  little  longer  than  the  thermometer  and  seal  at  one  end.  Fasten 
the  tube  firmly  to  the  thermometer  with  fine  wire  or  thread.  As  the 
thermometer  scale  is  to  be  used  for  measuring  the  length  of  the  liquid 
column,  the  lower  end  of  the  tube  must  not  be  below  the  bottom  of  the 
thermometer  scale.  The  measurement  will  be  simplified  by  placing  the 
end  of  the  bore  of  the  tube  exactly  at  the  zero  of  the  scale.  The  stirrer 
may  be  made  of  a  piece  of  wire  about  30  in.  long,  bent  into  a  close  flat 
coil  that  fits  loosely  into  the  jar ;  with  the  remainder  of  the  wire  at  right 
angles  to  the  coil  for  a  handle.  Alcohol,  kerosene,  olive  oil,  turpentine, 
and  ether  are  suitable  liquids  for  the  study  of  expansion.  ] 


94 


HEAT 


I.  To  find  the  expansion  (in  'com.}  of  1  ccm.  of  a 
liquid 1  for  1°  rise  of  temperature. 

a.  Fill  one  of  the  jars  about  half  full  of  crushed  ice; 
then  fill  with  water  full  enough  to  cover  the  liquids  in  the 

tubes  fastened  to  the 
thermometers.  Fill 
the  flask  about  half 
full  of  water,  and  sup- 
port it  on  the  ring 
stand  with  the  gauze 
under  it  (Fig.  35). 
The  gauze  should  be 
about  10  cm.  above  the 
burner,  and  the  flame 
adjusted  so  that  it 
does  not  burn  above 
the  gauze ;  it  is  liable 
to  break  the  flask  if 
it  does.  Insert  the 
stopper  and  delivery 
tube,  being  careful  to 
press  the  stopper  in  firmly.  Fill  the  other  two  hydrometer 
jars  with  water. 

In  the  experimenting  that  follows,  one  jar  is  to  be  used  for 
temperatures  between  0°  and  4°,  another  for  temperatures 
between  20°  and  40°,  and  the  third  for  temperatures  between 
50°  and  80°.  Each  set  of  jars  will  serve  for  more  than  one 
student. 

The  temperatures  of  the  jars  are  raised  by  passing  in  steam 
from  the  flask,  and  lowered  either  by  allowing  to  stand  or 
by  pouring  out  part  of  the  water  and  adding  cold  water.  The 
jars  must  not  be  subjected  to  large  and  sudden  changes  of  tem- 
perature or  they  will  break. 

1  Insert  the  name  of  the  liquid  used. 


COEFFICIENT   OF   EXPANSION   OF    LIQUIDS  95 

Turn  off  the  gas  when  not  using  steam,  and  immediately 
remove  the  delivery  tube  from  the  jar.  If  this  precaution  is 
not  observed,  the  water  in  the  jar  will  be  "sucked" 
over  into  the  flask  when  the  steam  in  the  latter 
cools. 

b.  Take  a  thermometer  and  tube  containing  any  of 
the  liquids  provided  except  water ;  and,  after  thor- 
oughly stirring  the  ice  water  by  moving  the  stirrer 
several  times  up  and  down  through  the  length  of  the 
jar,  place  the  thermometer  and  tube  in  it.     Read 
the  temperature  and  the  height  of  the  liquid  on  the 
thermometer  scale.     The  latter  must  be  read  as  accu- 
rately as  possible  to  a  tenth  of  a  degree  division, 
reading  the  bottom  of  the  curved  surface.     Be  careful 
not  to  cause  the  tube  to  slip  along  the  thermometer. 
Unless  securely  fastened,  it  may  slip  far  enough  to 
cause  a  very  large  error  in  determining  the  expansion, 
which  will  be  only  a  few  divisions. 

c.  Read  and  record  the  position  of  the  bottom  of 
the  liquid  column.     (The  tube  need  not  be  in  the 
water  when  this  reading  is  taken.) 

d.  Find  the  length  of  the  liquid  column,  measured 
in  degree  divisions  on  the  thermometer  scale.     Thus, 
if  the  bottom  is  at  — 10°  and  the  top  at  91.3°,  the 
length  is  101.3  (not  marked  with  the  degree  sign). 

e.  After  thoroughly  stirring  a  jar  of  hot  water, 
put  the  thermometer  and  tube  into  it.     The  hot  water 
must  be  below  the  boiling  point  of  the  liquid  in  the  tube; 
otherwise  it  will  boil.      If  you  are  experimenting 

with  ether  the  temperature  must  not  exceed  33°.  Test  by 
inserting  only  the  bulb  of  the  thermometer  at  first.  Ether 
must  be  kept  away  from  the  flame,  as  its  vapor  is  very  in- 
flammable. If  alcohol  is  used,  the  water  must  not  be  above 
7.V.  Read  the  temperature  and  the  height  of  the  liquid  as 
before. 


96  HEAT 

/  Since  the  bore  of  the  tube  is  uniform,  the  volume  of  the 
liquid  is  proportional  to  its  length.  In  fact  the  volume  of  the 
tube  between  two  adjacent  marks  on  the  thermometer  scale 
may  be  taken  as  the  unit  of  volume. 

The  observed  expansion  is  not  linear,  although  indicated  by 
an  increase  of  length,  but  cubical.  Why  ? 

Compute  the  average  expansion  per  degree  of  rise  of  temper- 
ature. 

g.  The  average  expansion  per  degree  is  what  fraction  of 
the  volume  at  the  lower  temperature  ?  This  is  the  average 
coefficient  of  (cubical)  expansion  of  the  liquid  between  the 
observed  temperatures. 

Compare  the  value  you  have  obtained  with  the  value  given 
in  Table  V  of  the  Appendix. 

II.  To  find  the  average  coefficient  of  expansion  of 
water  for  the  different  intervals  of  temperature  specified 
in  the  record. 

a.  Proceed  in  a  similar  manner  with  the  thermometer  and 
tube  of  water.     Eead  the  position  of  the  bottom  of  the  water 
column. 

b.  After  thorough  stirring,  take  the  temperature  and  read 
the  position  of  the  top  of  the  water  column  as  accurately  as 
possible:  (1)  in  a  jar  of  ice  water;  (2)  in  water  at  about  the 
temperature  of  the  laboratory;  (3)  in  water  at  45°  to  50°;  in 
water  at  75°  to  80°. 

c.  Compute  the  average  coefficient  of  expansion  of  water: 
(1)  between  the  first  and  second  observed  temperatures;  (2) 
between  the  first  and  third  observed  temperatures ;  (3)  between 
the  third  and  fourth  observed  temperatures. 

d.  What  do  your  results  indicate  in  regard  to  the  uniformity 
of  the  expansion  of  water  at  different  temperatures  ?     (See 
values  given  in  Table  V  of  the  Appendix.) 

What  can  you  say  of  the  accuracy  of  this  method  for  deter- 
mining the  expansion  of  water  below  20°  ? 


COEFFICIENT   OF   EXPANSION   OF   AIR  97 

EXERCISE   32.     COEFFICIENT   OF   EXPANSION 
OF    AIR 

References.  —  Hoadley,  268-270;  Carhart  and  Chute,  318- 
320. 

To  find  by  what  fraction  of  its  volume  at  0°  air  ex- 
pands when  Us  temperature  is  raised  1° 

Apparatus.  —  Copper  boiler  on  tripod,  with  tall  top ;  Bunsen 
burner;  hydrometer  jar;  stirrer;  thermometer,  with  attached 
tube  containing  air  and  a  mercury  index ;  ice. 

[The  tube  containing  the  air  must  be  of  small  bore  (1  mm.  or  less),  in 
order  to  hold  the  mercury  index  in  position,  and  should  be  10  or  12  in. 
long.  Prepare  as  follows  :  Thoroughly  dry  the  tube  by  passing  dried  air 
through  it.  Insert  an  end  of  the  tube  into  mercury,  withdraw  a  column 
about  3  mm.  long,  and  let  it  run  some  distance  down  the  tube.  Seal  an 
end  of  the  tube  in  a  flame  ;  fasten  the  tube  to  a  chemical  thermometer  as 
for  the  preceding  exercise  (see  directions).  Work  the  index  into  proper 
position  with  a  fine  wire,  allowing  for  an  expansion  of  somewhat  more 
than  one  fourth  without  exceeding  the  thermometer  scale.  Stick  a 
wooden  plug  in  the  end  (not  air  tight)  to  keep  out  moisture.] 

a.  The  tube  fastened  to  the  thermometer  contains  air  which 
is  confined  by  means  of  the  drop  of  mercury.  Any  considerable 
jarring  or  rough  handling  is  apt  to  displace  the  mercury  index 
and  vary  the  amount  of  air  confined  below  it.  If  this  happens, 
the  whole  set  of  observations  will  be  worthless.  With  any 
expansion  or  contraction  of  the  confined  air  the  index  changes 
its  position  without  permitting  any,  air  to  pass  it  (unless  it  is 
jarred).  The  length  of  the  air  column  is  to  be  measured  by 
the  thermometer  scale.  If  the  tube  extends  below  the  zero 
of  the  scale,  the  length  of  the  air  column  will  be  the  sum  of 
the  readings  of  the  lower  end  of  the  bore  of  the  tube  and  of  the 
lower  end  of  the  index.  For  example,  if  these  are  respectively 
—  12°  and  74.3°,  the  length  of  the  column  is  86.3  (not  marked 
with  the  degree  sign). 

COLEMAN'S  PHY.  LAB.  MAN.  —  7 


98  HEAT 

Without  handling  the  thermometer  or  lube  (to  avoid  impart- 
ing heat  from  the  hands)  read  the  lower  end  of  the  air  column, 
the  lower  end  of  the  index,  and  the  temperature,  all  to  .1°. 

b.  Fill  the  hydrometer  jar  half  full  of  crushed  ice,  and  pour 
in  water  enough  to  reach  the  top  of  the  scale  of  the  ther- 
mometer when  it  is  inserted.     While  the  water  is  cooling,  fill 
the  boiler  two  thirds  full  of  water,  and  heat  it  as  hot  as  you 
can  conveniently  bear  with  the  hand. 

Thoroughly  stir  the  water  in  the  jar,  pushing  the  ice  to  the 
bottom  of  the  jar  with  the  stirrer  several  times.  When  the 
temperature  has  fallen  to  1°  or  2°,  hold  the  ice  at  the  bottom 
of  the  jar  with  the  stirrer,  and  insert  the  thermometer  and  air 
tube.  Take  the  temperature  and  the  length  of  the  air  column 
as  before,  after  first  assuring  yourself  that  the  readings  of 
the  thermometer  and  of  the  mercury  index  have  become 
stationary. 

c.  Empty  the  jar  (into  the  supply  vessel  of  ice  water  if  one 
is  provided),  fill  it  with  water  from  the  faucet  and  empty  it 
several  times,  then  rinse  it  with  water  from  the  boiler  not 
hotter  than  you  can  comfortably  bear  with  the  hands.     (If 
thick  glass  is  heated  suddenly  it  will  break.)     Now  heat  the 
water  in  the  boiler  to  about  60°,  fill  the  jar  with  it,  and  insert 
the  thermometer  and  air  tube.     Put  the  top  on  the  boiler  and 
apply  heat  to  boil  the  remaining  water.     (The  boiler  should  be 
at  least  one  third  full.) 

After  thoroughly  stirring  the  water  in  the  jar,  take  the  tem- 
perature and  the  position  of  the  index  as  soon  as  they  have 
become  stationary. 

d.  Insert  the  thermometer  and  tube  into   the   top  of  the 
boiler.     Hold  it  by  the  cord  and  be  careful  not  to  burn  your- 
self with  the  steam,  which  must  be  escaping  freely  from  the 
top.     As  soon  as  the  thermometer  and  index  are  stationary, 
take  a  set  of  readings  as  before. 

Empty  the  water  from  the  jar  and  leave  the  thermometer 
and  tube  in  it. 


COEFFICIENT   OF   EXPANSION   OF   AIR  99 

Discussion.  —  a.  Compute  to  three  decimal  places  the  con- 
traction of  the  air  column  per  degree  fall  of  temperature  be- 
tween the  first  and  second  temperatures. 

b.  Compute  the  expansion  of  the  air  column  per  degree  rise 
of  temperature  between  the  first  and  third  temperatures. 

c.  Compute  the  same  between  the  third  and  fourth  tem- 
peratures. 

d.  Except  for  experimental  errors  the  change  of  length  per 
degree  should  be  the  same  by  the  three  computations.     (The 
expansion  of  gases  is  uniform.)     How  great  is  the  per  cent  of 
difference  between  your  results  ? 

e.  Take  the  average  of  your  three  values  of  the  expansion 
(or  contraction)  per  degree  change  of  temperature.     This  is  the 
most  reliable  value  obtainable  from  your  observations. 

/.  Assuming  the  same  contraction  per  degree,  compute  the 
length  of  the  air  column  at  0°. 

g.  The  expansion  per  degree  (e)  is  what  fraction  of  the 
length  at  0°  ?  This  is  the  coefficient  of  expansion  of  air  (and 
of  all  gases).  Its  true  value  is  .00366.  Compute  the  per  cent 
of  error  of  your  result. 

EXEECISE   33.     MELTING   AND  FKEEZING. 
SOLUTION 

References.  — Hoadley,  271-273;  Carhart  and  Chute,  329- 
332  and  334;  Sanford,  pp.  152-154 ;  Slate,  123-124 ;  Jones,  45; 
Madan,  pp.  39-42,  150-153,  and  158-159. 

Apparatus.  —  Thermometer,  numbered  for  identification  ; 
tumbler  or  beaker  ;  test  tube  ;  access  to  ice  and  salt. 

[It  is  suggested  that  all  the  thermometers  used  by  the  students  be 
numbered,  and  that  a  table  of  corrections  for  their  freezing  and  boiling 
points  be  posted  in  the  laboratory  for  convenient  reference.  Such  a 
table  may  be  compiled  from  the  records  of  this  exercise  and  Exercise  35  ; 
and,  when  once  compiled,  will  serve  as  a  check  on  the  work  of  future 
classes.  ] 


100  HEAT 

I.  To  find  the  correction  for  the  melting  point  of  a 
thermometer. 

a.  Fill  the  tumbler  about  half  full  of  fine  crushed  ice.     In- 
sert the  thermometer,  and  pack  the  ice  about  it  nearly  to  the 
zero  of  the  scale.     After  the  mercury  becomes  stationary,  read 
the  temperature  accurately  to  .1°.     Record  the  number  of  the 
thermometer  and  the  reading  in  melting  ice.     The  graduation 
of  most  thermometers,  except  expensive  ones,  is  appreciably  in- 
accurate.    In  melting  ice  the  reading  should  be  exactly  zero. 
The  reading  you  obtained  is  therefore  the  error  of  the  melting 
point  of  this  thermometer. 

b.  What  evidence  is  there  that  the  ice  you  used  was  melting  ? 
Was  the  ice  receiving  or  losing  heat  during  the  experiment  ? 

Give  reason  for  your  opinion. 

II.    To  find  whether  ice  freezes  and  melts  at  the  same 
temperature. 

a.  Mix  with  the  ice  about  one  third  its  volume  of  table  salt. 
Put  enough  water  into  the  test  tube  to  fill  it  about  one  fourth 
full  after  the  thermometer  is  inserted,  and  place  it  in  the  freez- 
ing mixture  of  salt  and  ice.     Stir  the  mixture  with  the  test 
tube,  keeping  watch  of  the  temperature  of  the  water  in  it.     At 
what  temperature  does  it  freeze  ?     Sometimes  the  temperature 
of  water  falls  a  few  degrees  below  the  freezing  point  before  it 
begins  to  freeze  ;  but  as  soon  as  freezing  begins,  the  tempera- 
ture very  quickly  rises  to  the  freezing  point  and  remains  sta- 
tionary till  the  process  is  completed.     Observe  -whether  this 
happens  in  your  experiment.     Read  accurately  and  record  the 
freezing  point. 

b.  Does  the  temperature  fall  below  the  freezing  point  after 
the  water  is  all  frozen  ? 

To  melt  the  ice  in  the  test  tube  let  water  run  on  it  from  the 
faucet.  Insert  the  thermometer  into  the  freezing  mixture  and 
take  its  temperature. 


MELTING   AND   FREEZING.    SOLUTION  101 

Was  the  water  in  the  test  tube  receiving  or  losing  heat  during 
the  experiment  ?  Give  reasons  for  your  answer. 

c.  How  do  the  temperatures  of  melting  ice  and  freezing 
water  compare  ? 

What  determines  whether,  in  a  mixture  of  the  two,  the  ice 
will  melt  or  the  water  freeze  ? 

III.  To  observe  the  effect  of  pressure  on  the  melting 
point  of  ice. 

Apparatus.  —  A  block  of  ice  supported  at  the  ends  ;  a  heavy 
weight  suspended  from  the  block  of  ice  by  means  of  a  loop  of 
fine  wire  passed  over  it. 

a.  When  the  weight  was  hung  upon  the  ice,  the  wire  rested 
upon  its  surface.     How  do  you  find  it  now  ?     Look  at  it  from 
time  to  time  during  the  hour  and  note  any  change  in  the  posi- 
tion of  the  wire. 

b.  How  is  the  cut  that  the  wire  makes  in  the  ice  mended  ? 
What  is  the  cause  of  the  melting  under  the  wire  ? 

What  is  the  source  of  the  heat  required  for  this  melting  ? 
Why  does  the  water  above  the  wire  freeze  ? 

IV,  To  observe  the  effect  on  temperature  of  dissolving 
ammonium  chloride  or  ammonium  nitrate  in  water. 

Apparatus.  —  Thermometer  ;  test  tube  ;  ammonium  nitrate  or 
ammonium  chloride. 

a.  Fill  the  test  tube  about  one  third  full  of  water  and  take 
its  temperature.     Add  a  teaspoonful  or  more  of  ammonium 
nitrate  or  ammonium  chloride,  stir  with  the  thermometer,  and 
note  the  change  of  temperature. 

b.  What  inference  may  be  drawn  from  this  change  of  tem- 
perature ? 

What  points  of  similarity  are  there  between  solution  and 
melting  ?  What  transformation  of  heat  is  involved  in  either 
process  ?  (See  references.) 


102  HEAT 

EXERCISE  34.     EVAPORATION:     VAPOR  PRESSURE. 
DEW-POINT 

References.  —  Hoadley,  274-276;  Carhart  and  Chute,  335- 
336  and  339-341 ;  Slate,  125,  128-129,  and  133-135 ;  Sanford, 
pp.  162-165  ;  Jones,  48-49,  59-63,  and  95-96;  Madan,  pp.  165- 
168,  220-223,  and  342-348. 

I.  To  compare  the  evaporation  of  water,  alcohol,  and 
ether  with  respect  to  rapidity  and  effect  upon  tempera- 
ture. 

Apparatus.  —  Water ;  small  bottles  of  alcohol  and  ether. 

a.  Wet  the  palm  with  water  and  move  the  hand  rapidly 
back  and  forth  edgewise.      What  is  the  temperature  sensation? 
Explain  it. 

b.  Repeat  with  alcohol.     Wet  the  palm  by  placing  it  over 
the  mouth  of  the  bottle  and  inverting  the  bottle.     Compare  the 
rapidity  of   evaporation  of  alcohol  and  water.     Compare  the 
temperature  sensations.     Account  for  the  difference. 

c.  Again  moisten  the  palm  with  alcohol,  and  compare  the 
temperature  sensations  when  the  hand  is  still  and  when  it  is 
moved  rapidly  to  and  fro.     Account  for  the  difference. 

d.  Repeat  with  ether.     Compare  results  with  those  obtained 
with  alcohol.     Account  for  the  difference. 

II.  To  find  the  pressure  of  saturated  vapor  of  water, 
alcohol,  and  ether  at  the  temperature  of  the  ldboratoj*y ; 
and  to  observe  the  effect  of  change  of  temperature  on  tlxe 
pressure  of  a  saturated  vapor. 

Apparatus.  —  Three  barometer  tubes  set  up,  one  with  water, 
one  with  alcohol,  and  one  with  ether  in  the  tube  above  the 
mercury,  each  supported  with  ring  stand  and  clamp ;  meter 
rod;  access  to  a  barometer. 

[To  set  up  a  tube  fill  it  up  with  clean  mercury  within  an  inch  or  less 
of  being  full,  then  finish  filling  with  the  liquid  (water,  alcohol,  or  ether). 


EVAPORATION.    VAPOR   PRESSURE.     DEW-POINT      103 

Invert  a  few  times  to  gather  up  air  bubbles  in  the  mercury  by  means  of  the 
lighter  liquid  as  it  runs  from  end  to  end.  Fill  the  tube  again  completely 
full  and  invert  it  over  a  dish  of  mercury.  If  all  air  has  been  removed, 
the  liquids  will  rise  and  completely  till  the  tubes  when  they  are  inclined.] 

a.  Read  the  laboratory  barometer  and  thermometer. 

b.  Measure  the  height  of  the  mercury  columns  in  the  tubes 
with  water,  alcohol,  and  ether  respectively  at  the  top. 

c.  Unclamp  the  tube  containing  alcohol  or  ether,  and  incline 
it,  being  careful  to  hold  the  lower  end  firmly  under  the  surface 
of  the  mercury  in  the  dish.     Incline  it  till  the  space  at  the  top 
entirely  disappears.    Clamp  the  tube  in  a  vertical  position  again. 

What  evidence  is  there  that  the  space  above  the  liquid  does 
not  contain  any  air  ? 

What  evidence  is  there  that  it  is  not  a  vacuum  ?  It  does 
contain  the  vapor  of  the  liquid  above  the  mercury. 

How  high  would  the  mercury  stand  in  the  tube  if  the  vapor 
were  not  present  ? 

What  became  of  this  vapor  when  the  tube  was  inclined  ? 

d.  How  great  a  pressure  (measured  in  centimeters  of  mer- 
cury) is  exerted  by  the  vapors  of  water,  alcohol,  and  ether  at 
the  temperature  of  the  room  ? 

e.  Warm  the  tube  containing  the  ether  by  holding  the  hands 
on  it,  and  observe  the  effect  on  the  height  of  the  mercury. 
(No  measurements  need   be  taken.)      The   observed  effect  is 
partly  due  to  the  heating  of  the  vapor  already  in  the  tube, 
but  chiefly  to  the  evaporation  of  more  ether.     So  long  as  any 
liquid  ether  remains  it  will  continue  to  evaporate  till  the  space 
above  it  is  saturated;  that  is,  contains  all  the  ether  vapor  that 
it  can  hold  at  that  temperature. 

Is  the  pressure  of  saturated  ether  vapor  greater  or  less  at 
a  higher  temperature  ? 

/.  Try  the  effect  of  warming  the  other  tubes  with  the  hands. 
State  results  and  compare  with  that  obtained,  with  ether. 

g.  Point  out  the  agreement  between  the  results  of  these 
experiments  and  those  on  evaporation. 


104  HEAT 

III.    To  find  the  dew-point  of  the  air  in  the  laboratory. 
Apparatus.  —  Thermometer j    bright  calorimeter  or  tin  can; 

two   beakers   or  tumblers;    ice,    or   ammonium   nitrate   or  ammo- 
nium chloride. 

a.  Fill  the  calorimeter  with  water  to  about  an  inch  in  depth. 
Have  at  hand  a  tumbler  of  water  and  a  little  finely  crushed  ice 
in  the  other  tumbler.     Add  ice  to  the  calorimeter,  a  very  little 
at  a  time,  stirring  constantly  with  the  thermometer.     Watch 
closely  meanwhile  for  the  first  deposit  of  moisture  on  the  calo- 
rimeter near  the  bottom ;  and,  when  it  appears,  take  the  tem- 
perature of  the  water. 

If  the  dew-point  is  below  0°,  it  will  be  necessary  to  add  salt 
with  the  ice.  If  the  calorimeter  becomes  filled  to  a  depth 
of  over  2  in.,  pour  out  part  of  the  contents. 

Wipe  the  dew  off  with  a  cloth  or  your  finger,  and  observe 
whether  it  quickly  gathers  again.  If  it  does,  warm  the 
water  in  the  calorimeter  slightly  by  pouring  in  a  little  water 
from  the  tumbler.  Try  to  find  the  highest  temperature 
at  which  the  dew  will  form  at  all.  The  warm,  moist  breath 
will  .cause  the  dew  to  form,  on  the  side  of  the  calorimeter 
toward  you  before  it  will  elsewhere.  Test  this  by  holding 
the  mouth  close  to  the  calorimeter  and  breathing  against  it. 
To  get  the  dew-point  of  the  air  in  the  room,  avoid  this  error 
by  turning  the  calorimeter  and  quickly  observing  the  appear- 
ance of  the  farther  side. 

b.  Wipe  the  calorimeter ;  and,  starting  with  the  temperature 
low  enough  to-  deposit  a  thin  film  of  dew,  stir  constantly  till 
it  disappears,  then  quickly  note  the  temperature.     The  tem- 
peratures at  which  the  dew  appears  and  disappears  should  not 
differ  by  more  than  1°.     Take  their  average  as  the  dew-point 
of  the  air  in  the  laboratory  at  the  time  of  the  experiment. 

NOTE.  —  Water,  taken  at  the  temperature  of  the  laboratory,  can  gen- 
erally be  lowered  to  the  dew-point  by  dissolving  ammonium  nitrate  or 
ammonium  chloride  in  it.  If  ice  is  not  provided,  use  one  of  these  salts 
instead,  adding  it  slowly  while  stirring  with  the  thermometer. 


BOILING   OF    WATER  105 

EXERCISE  35.     BOILING   OF   WATER 

References.  —  Hoadley,  277-280;  Carhart  and  Chute,  337- 
338. 

Apparatus.  —  Chemical  thermometer,  numbered  for  identifi- 
cation ;  ring  stand,  ring,  and  clamp ;  wire  gauze ;  large  flask, 
and  stopper  to  fit,  with  two  holes ;  delivery  tube  (Fig.  35) ; 
hydrometer  jar ;  Bunsen  burner ;  closed  tube  pressure  gauge 
containing  water  in  the  closed  tube  above  the  mercury  (Fig.  37). 

[To  make  the  pressure  gauge  take  a  piece  of  small  glass  tubing  about 
a  foot  long ;  seal  one  end  ;  and  bend  about  3  in.  from  the  closed  end,  as 
shown  in  Fig.  41,  making  the  bend  narrow  enough  to  pass  through  the 
neck  of  the  flask.  Pour  in  enough  mercury  to  fill  the  short  arm  and 
extend  just  past  the  bend.  By  holding  the  tube  horizontal,  with  the 
closed  end  below,  and  tilting  first  one  end,  then  the  other,  the  air  can 
be  gradually  displaced  from  the  closed  tube  by  the  mercury.  Next  pour 
in  water  to  a  depth  of  about  half  an  inch,  and  work  a  little  of  it  into  the 
closed  arm  by  inclining  the  tube  with  the  closed  arm  above.  ] 

I.  To  observe  the  phenomena  preceding  and  accom- 
panying boiling;  and  to  find  the  temperature  of  the 
boiling  water  and  the  steam. 

a.  Fill  the  flask  about  half  full  of  water  from  the  faucet, 
wipe  the  outside  dry,  place  it  on  the  wire  gauze  on  the  stand 
and  apply  heat.  The  flame  must  not  be  high  enough  to  burn 
above  the  gauze.  Insert  the  thermometer  into  the  flask,  and 
occasionally  observe  the  temperature.  Observe  the  water  care- 
fully from  the  moment  when  you  begin  to  apply  the  heat,  and 
note  the  first  formation  of  bubbles.  They  are  bubbles  of  air 
which  were  dissolved  in  the  water  and  are  now  being  driven 
off  by  the  heat.  Describe  their  size,  abundance,  and  behavior; 
and  state  through  what  range  of  temperature  (approximately) 
they  continue  to  be  given  off. 

6.  Note  any  gathering  of  moisture  on  the  inside  of  the  flask 
and  on  the  thermometer.  Does  it  occur  before  the  water  boils  ? 
How  do  you  account  for  it  ? 


106  HEAT 

c.  Note  the  temperature  when  sounds  begin  to  come  from 
the  flask.  What  is  their  cause  ?  Is  the  water  boiling  when 
the  sounds  begin  ?  Watch  closely  for  the  first  formation  of 
bubbles  larger  than  the  air  bubbles  first  observed.  What  are 
they?  Where  are  they  formed?  What  becomes  of  them? 
Note  the  temperature.  Watch  closely  for  any  change  in  the 
phenomena  as  the  temperature  approaches  100°. 

d.  Cause  the.  water  to  boil  slowly  and  note  the  tem- 
perature.   Boil  rapidly  and  again  note  the  temperature. 

e.  Raise  the  thermometer  till  the  bulb  is  just  out 
of  the  water.    What  is  the  reading  of  the  thermometer  ? 
Read  this  as  accurately  as  possible  and  record  it  as  the 
temperature  of  the  steam. 

II.  To  find  the  vapor  pressure  of  steam  at  the 
boiling  point. 

a.  The  closed  tube  pressure  gauge  (Fig.  37)  contains 
water  in  the  closed  arm  above  the  mercury.     Lower  it 
into  the  steam  above  the  boiling  water  in  the  flask, 
and  note  the  formation  of  water  vapor  in  the  closed 
arm.    Observe  the  level  of  the  mercury  in  the  two  arms. 

How  does  the  pressure  of  the  water  vapor  in  the 
closed  arm  compare  with  the  atmospheric  pressure  ? 
What  is  its  temperature  ? 

b.  Observe  the  effect  of  removing  the  tube  from  the 
flask.     Explain. 

c.  What  determines  the  temperature  at  which  any 
FIG.  37.    liquid  boils? 

III.  To  observe  the  effect  of  increase  of  pressure  upon 
the  boiling  point;  and  to  find  the  correction  for  the 
boiling  point  of  the  thermometer  used. 

(L  Remove  the  burner  from  under  the  flask  while  you  are 
making  the  following  adjustment.  Push  the  thermometer 
through  the  hole  in  the  stopper  till  the  bulb  is  but  little  above 


BOILING   OF    WATER  107 

the  water  when  the  stopper  is  in  the  flask.  Be  careful  not  to 
break  the  thermometer  by  prying  or  using  too  much  force.  If 
you  have  difficulty,  call  for  assistance.  Press  the  stopper 
firmly  into  the  flask.  Connect  the  delivery  tube  as  shown  in 
Fig.  35,  and  let  it  extend  to  the  bottom  of  the  hydrometer  jar, 
which  should  be  nearly  full  of  water.  Boil  the  water  in  the 
flask  and  take  the  temperature  of  the  steam.  Gradually  raise 
the  delivery  tube  out  of  the  jar  while  observing  the  effect  upon 
the  temperature  of  the  steam.  Raise  and  lower  the  delivery 
tube  till  you  are  sure  of  the  effect. 

b.  What  change  in  the  atmospheric  pressure  would  produce 
the  same  effect  upon  the  temperature  of  the  steam  as  lowering 
the  delivery  tube  into  the  jar  of  water?     Estimate  roughly  the 
change  in  the  barometer  corresponding  to  the  water  pressure 
in  the  bottom  of  the  jar. 

Empty  the  flask  and  return  the  thermometer  to  its  case. 

c.  Head  the  barometer. 

d.  At  a  pressure  of  one  atmosphere  (76  cm.)  the  true  value 
of   the   boiling  point  is    100°.     For   pressures  either  *  slightly 
greater  or  less  than  one  atmosphere,  the  boiling  point  varies 
.37°  for  a  change  of  pressure  of  1  cm.     Compute  the  true  value 
of  the  boiling  point  at  the  observed  barometric  pressure. 

e.  What  is  the  error  of  the  boiling  point  of  this  thermometer  ? 

/.  You  will  find  in  text-books  a  description  of  better  appara- 
tus for  the  accurate  location  of  the  boiling  point  on  a  thermom- 
eter. Will  the  boiling  point  as  determined  by  this  apparatus 
be  a  very  little  too  high  or  too  low  ?  Why  ? 

EXERCISE   36.     THE  BOILING   POINT  OF  A  LIQUID 
(DETERMINED  BY  MEANS  OF  ITS  VAPOR  PRESSURE) 

Apparatus. — Flask  fitted  with  stopper  and  delivery  tube 
and  supported  on  a  ring  stand  (Fig  35) ;  Bunsen  burner ;  hy- 
drometer jar;  stirrer;  thermometer;  pressure  gauges  contain- 
ing the  liquids  whose  boiling  points  are  to  be  determined. 


108  HEAT 

[The  pressure  gauges  are  like  that  of  the, preceding  exercise  except  that 
the  closed  arm  should  be  8  or  9  in.  long,  and  the  open  arm  a  foot  or 
more.  See  the  preceding  exercise  for  directions  for  filling.  Any  liquid 
whose  boiling  point  lies  between  30°  and  90°  may  be  used.  ] 

I.  To  find  the  boiling  point  of • 

a.  Fill  the  flask  about  half  full  of  water,  and  adjust  the 
apparatus  as  shown  in  (Fig.  35),  being  careful  to  press  the 
stopper  in  firmly .    Apply  heat.    Fill  the  hydrometer  jar  nearly 
full  of  water,  and  place  in  it  the  pressure  gauge  containing  the 
liquid  whose  boiling  point  is  to  be  determined.     The  water 
must  cover  the  closed  tube  of  the  gauge.     Pass  steam  into  the 
jar  till  the  mercury  stands  at  the  same  level  in  the  arms  of  the 
gauge.     Remove  the  flame  from  under  the  flask  and  the  delivery 
tube  from  the  jar  ;    and  stir  the  water  in  the  jar  thoroughly. 
If  the  mercury  levels  in  the  gauge  now  indicate  that  the  water 
is  below  the  boiling  point  of  the  liquid,  pass  in  a  little  more 
steam.     If  the  temperature  is  too  high,  allow  to  cool,  stirring 
occasionally,  till  the  mercury  stands  exactly  at  the  same  level 
in  the  two  arms ;   then  take  the  temperature,  calling  it  the 
boiling  point  of  the  liquid. 

b.  How  do  you  know  that  this  is  the  boiling  point  of  the 
liquid  ? 

II.  To  find  the  boiling  point  of . 

In  the  same  way  find  the  boiling  point  of  another  liquid. 

CALORIMETRY 

37.  The  Heat  Unit.  —  In  the  following  experiments  it  is 
required  to  measure  definitely  (although  indirectly)  the  quan- 
tity of  heat  transferred  from  one  body  to  another.  For  this 
purpose  it  is  necessary  to  adopt  a  definite  quantity  of  heat  as 
the  unit  of  measurement.  The  one  that  we  shall  use  is  the 
quantity  of  heat  required  to  raise  the  temperature  of  1  g.  of 
water  1°  C. ;  and  it  is  called  the  calorie. 


Thus,  to  raise^fche  temperature  of  20  g.  of  water  from  5°  to 
12°  would  require  140  calories ;  for  the  rise  of  temperature  is 
7°,  which  would  require  7  calories  per  grain,  or  20  x  7  calories 
for  20  g. 

38.  Specific   Heat.  —  Equal   quantities  of  heat   are   not  re- 
quired to  raise  the  temperature  of  equal  weights  of  different 
substances  the  same  number  of  degrees.     For  example,  it  is 
found  that  only  one  ninth  as  much  heat  is  required  to  raise  a 
given  weight  of  iron  1°  as  is  necessary  to  raise  the  same  weight 
of  water  1°.     This  is  expressed  by  saying  that  the  specific  heat 
of  iron  is  one  ninth  or  .11.     It  will  be  seen  that  specific  heat 
(like   specific   gravity)  is  a  ratio.     If,  however,  we  take  for 
comparison  1  g.  of   each  of  the   substances,  it  follows  that, 
since  it  takes  one  calorie  to  raise  1  g.  of  water  1°,  it  will  take 
one  ninth  of  a  calorie  to  raise  1  g.  of  iron  1°.     Hence  the  specific 
heat  of  a  substance  may  be  defined  as  the  number  of  calories 
of  heat   necessary  to  raise   the   temperature  of  1   g.  of  the 
substance  1°  C. 

To  illustrate :  If  the  specific  heat  of  a  substance  is  .04 
(calories),  to  raise  the  temperature  of  50  g.  of  it  from  2°  to  6° 
would  require  50  x  4  x  .04,  or  8  calories.  The  same  body  in  cool- 
ing from  50°  to  30°  would  give  out  50  x  20  x  .04,  or  40  calories. 

39.  The    Heat    Equation.  —  In    experiments   in   calorimetry 
(heat  measurement)   two  bodies  at  different  temperatures  are 
brought  together  in   the    same  vessel   (the   calorimeter),  and 
mixed  so  that  their  temperatures  are  quickly  equalized.     The 
calorimeter,  of  course,  shares  in  the  transfer  of  heat.     In  addi- 
tion to  this,  there  will  be  a  transfer  of  heat  to  or  from  bodies 
outside  the  calorimeter  by  conduction  and  radiation ;  and  for 
accurate  results  allowance  must  be  made  for  this,  or  the  con- 
ditions of  the  experiment  must  be  so  adjusted  that  the  gains 
and  losses  of  heat  by  these  processes  balance  each  other.     The 
latter  method  is  adopted  in  the  following  experiments.     (How 
this  is  accomplished  will  be  left  for  consideration  in  the  ex- 


110  UK AT 

periments  themselves.)  By  this  method  it  is  assumed  that 
the  transfers  of  heat  take  place  only  among  the  calorimeter  and 
its  contents ;  hence  it  follows  that  the  heat  given  out  by  the 
body  or  bodies  that  fall  in  temperature  is  equal  to  the  heat 
gained  by  the  bodies  that  rise  in  temperature. 

For  example,  let  it  be  required  to  find  the  specific  heat  (s) 
of  iron  from  the  following  data  :  — 

A  brass  calorimeter  weighing  100  g.  contains  400  g.  of  water 
at  18°.  Into  this  is  put  a  roll  of  iron  at  a  temperature  of  100° 
and  weighing  190  g.  The  temperature  of  the  calorimeter  and 
water  rises  and  that  of  the  iron  falls  to  22°.  It  is  further 
given  that  the  specific  heat  of  brass  (the  material  of  the  calo- 
rimeter) is  .09. 

SOLUTION 

Rise  of  temp,  of  calorimeter  and  water  =  22°—  18°  =  4° 
Gain  of  heat  by  calorimeter  =100  x  4  x  .09  =  36  cal. 

Gain  of  heat  by  water  =  400  x  4  =  1600  cal. 

Fall  of  temperature  of  iron  =  100°  -  22°  =  78° 

Loss  of  heat  by  iron  =  190  x  78  x  *  =  14820 seal. 

Loss  of  heat  by  iron  =  gain  of  heat  by  calorimeter 
+  gain  of  heat  by  water ; 
14820s  =  36 +  1600 
Specific  heat  of  iron  (*)=  1636 -s- 14820  =  .110 

The  method  of  treating  the  experimental  data,  as  illustrated 
by  the  above  example,  may  be  stated  as  follows  :  — 

(1)  Find  numerical  or  algebraic  expressions  for  the  separate 
quantities  of  heat  involved  in  the  equalization  of  temperatures. 

(2)  With  these  heat  quantities  form  the  heat  equation,  which 
expresses  the  equality  of  heat  lost  and  heat  gained.     This  equa- 
tion  contains    as  an   unknown  quantity  the  quantity  sought 
(specific  or   latent  heat) ;   and  this  is   found  by  solving   the 
equation  by  the  usual  algebraic  processes. 


OF  THE 

UNIVERSITY 

OF  i  SPECIFIC   HEAT  111 


EXERCISE  37.     SPECIFIC  HEAT 

References.  —  Hoadley,  281-283  ;  Carhart  and  Chute,  325- 
327  ;  Slate,  113. 

To  find  the  number  of  calories  given  out  by  1  g.  of 
brass  or  copper  in  cooling  1°. 

Apparatus.  —  Bunsen  burner  ;  copper  vessel  on  tripod,  or 
other  open  vessel  for  boiling  water  ;  roll  of  copper  or  brass 
with  fine  wire  attached  for  handle  ;  brass  or  copper  calorim- 
eter; thermometer;  platform  balance  and  weights  ;  mop  cloth. 

[The  roll  of  copper  or  brass  should  weigh  from  200  to  400  g.  ,  and  must 
be  an  open  roll  with  a  space  of  about  one  fourth  in.  between  the  sur- 
faces, to  avoid  holding  water  by  capillary  action  (Fig.  38).  It  would 
better  be  made  from  sheet  metal  at  least  1  mm.  thick.  ] 

a.  Begin  heating  the  water  in  the 
copper  boiler.    Weigh  the  roll  of  copper 
(or  brass)  to  .1  g. 

b.  Weigh  the  calorimeter.     Put  the 
roll  into  the  calorimeter  and  pour  in 
enough  water  to  cover  it.     The  water 
should  be  2°  or  3°  below  the  tempera- 
ture of  the  room  for  best  results.     Put 
the  roll  into  the  water  that  is  being 
heated,  and  see  that  there  is  enough 

water  in  the  boiler  to  cover  it.  FIG.  38. 

c.  Weigh  the  calorimeter  and  the  water  in  it,  and  remove 
it  from  the  balance. 

d.  After  the  above  has  been  done  and  the  water  in  the  copper 
vessel  is  boiling,  thoroughly  stir  the  water  in  the  calorimeter 
with  the  thermometer  and  take  its  temperature  to  .1°.    Kemove 
the  thermometer,  hold  the  calorimeter  close  beside  the  boiler, 
and  as  quickly  as  possible  transfer  the  roll  to  the  calorimeter. 
Place  the  calorimeter  on  the  table  at  a  distance  from  the  flame; 
move  the  roll  about  in  it  to  stir  the  water;  insert  the  ther- 


112  HEAT 

mometer  and  take  the  temperature  near  the  top  and  the  bottom 
of  the  water  and  on  opposite  sides  of  the  roll.  If  differences 
are  found,  stir  again. 

Record  the  highest  uniform  temperature. 

NOTE.  —  It  is  assumed  that  the  roll  is  at  a  temperature  of  100°  when 
put  into  the  calorimeter  ;  but  it  cools  with  great  rapidity  during  the  trans- 
fer, and  a  delay  of  a  second  in  this  process  will  cause  an  error  of  from 
5%  to  10%  in  the  result.  The  small  quantity  of  hot  water  clinging  to  the 
roll  and  transferred  with  it  partly  compensates  for  the  loss  of  heat  by  the 
roll.  The  temperatures  must  be  read  as  accurately  as  possible.  An  error 
of  .1°  in  determining  a  temperature  change  of  5°  is  an  error  of  2  %. 

It  will  be  well,  if  time  permits,  to  repeat  the  experiment.  It 
will  take  but  a  few  minutes  to  do  so;  and,  having  become 
familiar  with  the  method  of  procedure,  you  will  very  probably 
secure  better  results. 

The  specific  heat  of  copper  and  of  brass  are  equal  within  the 
limits  of  your  experimental  errors ;  hence  the  roll  and  the 
calorimeter,  either  of  which  may  be  of  brass  or  copper,  are 
considered  together  in  this  experiment,  and  their  specific  heat 
(s)  determined.  Remember  that  s  denotes  the  number  of  calories 
of  heat  lost  by  1  g.  of  either  metal  when  its  temperature  falls  1° 
and  the  number  of  calories  gained  when  its  temperature  rises  1°. 

The  heat  that  caused  the  rise  of  temperature  of  the  calo- 
rimeter and  water  is  assumed  to  come  entirely  from  the  roll, 
and  hence  to  be  equal  to  the  heat  lost  by  it.  Form  the  heat 
equation  with  these  equal  quantities  of  heat  and  solve  it  for  s. 
Include  the  equation  and  its  solution  in  the  record. 

OBSERVATIONS 

a.  Weight  of  copper  (or  brass)  roll  = g. 

b.  Weight  of  calorimeter  = g. 

c.  Weight  of  calorimeter  and  water  = g. 

d.  Initial  temperature  of  water  and  calorimeter  =  —     -°  C. 
Initial  temperature  of  copper  roll                    =    100°  C. 
Final  temperature  of  calorimeter  and  contents  = °  C. 


LATENT   HEAT   OF   FUSION   OF   ICE  113 

COMPUTATIONS 

Weight  of  water  = g. 

Rise  of  temperature  of  calorimeter 

Gain  of  heat  by  calorimeter  =  (      )  x  (  )  X  s= calories 

Fall  of  temperature  of  roll  =  — 

Loss  of  heat  by  roll  =(     )x(  )  x  s— calories 

Specific  heat  of  copper  or  brass  (s)  = 

To  find  the  specific  heat  of  any  metal. 

Use  a  roll  of  the  metal  as  above.  Instead  of  the  brass  calo- 
rimeter a  small  tin  can  or  glass  beaker  may  be  used.  In 
any  case  the  specific  heat  of  the  calorimeter  must  be  known. 
(See  Table  of  Specific  Heats  in  the  Appendix.) 

Follow  the  above  directions  and  form  of  record ;  but  in  the 
expression  for  the  gain  of  heat  by  the  calorimeter  substitute  its 
known  specific  heat. 

EXERCISE  38.     LATENT   HEAT  OF   FUSION  OF  ICE 

References.  — Hoadley,  284;  Carhart  and  Chute,  329-333; 
Sanford,  pp.  152-153. 

To  find  the  number  of  calories  required  to  change  1.  g. 
of  ice  at  0°  into  water  at  0°. 

Apparatus.  —  Calorimeter;  thermometer  and  stirrer  (Fig.  39); 
platform  balance  and  weights ;  cloth ;  supply  of  ice  and  of  hot 
water. 

[For  a  stirrer  use  a  piece  of  very  thin  copper  (thickness  of  writing 
paper)  about  1  x  1.5  in.  with  two  holes  large  enough  to  slip  it  on  the  end 
of  the  thermometer.  ] 

a.    Weigh  the  calorimeter  to  .1  g. 

6.  Fill  the  calorimeter  about  half  full  of  water  (about  200  g.) 
at  about  45°  C.  Take  hot  water  from  the  supply  and  add  cold 
water  as  needed.  Weigh  the  calorimeter  and  water  and  remove 
from  the  balance. 

COLEMAN'S  PHY.  LAB.  MAN.  —  8 


114  HEAT 

c.  Have  at  hand  a  large  handful  of  crushed  ice  on  a  cloth 
(to  keep  it  dry).     There  should  be  no  pieces  larger  than  small 

marbles.  Stir  the  water  with  the  copper  stirrer  on 
the  thermometer  till  the  temperature  is  uniform, 
then  take  it  accurately.  Quickly  dry  the  ice  by 
spreading  it  out  thin  and  wiping  it  with  the  cloth, 
and  immediately  put  nearly  all  of  it  into  the  calo- 
rimeter. Stir  the  water  constantly  while  the  ice  is 
melting.  Keep  the  hands  off  the  calorimeter  during 
the  process  to  avoid  conduction  from  the  hand. 

If  ice  remains  after  the  temperature  has  fallen  to 
8°  or  10°,  remove  it  with  the  stirrer.  If  the  ice  is  all 
melted  before  the  temperature  has  fallen  to  10°  or 
12°,  add  more  without  delay. 

As  soon  as  the  ice  has  all  disappeared,  stir  the 
water  thoroughly  and  take  the  temperature  at  top 
and  bottom.  If  there  is  a  difference,  stir  and  read 
again.  Record  the  lowest  uniform  temperature  of 
the  water. 

d.  Weigh  the  calorimeter  and  contents. 

It  will  be  advisable  to  repeat  the  experiment  if  there  is  time. 
Do  not  leave  ice  to  melt  on  the  table.  Leave  the  calorimeter 
empty  and  the  table  dry. 

e.  The  experiment  is  planned  so  that  the  loss  of  heat  from 
the  calorimeter  and  contents  by  radiation  and  conduction  in  the 

.  first  part  of  the  experiment  is  approximately  balanced  by  the 
gain  by  the  same  means  in  the  latter  part.  Hence  the  heat 
necessary  to  melt  the  ice  and  to  raise  its  temperature  after 
melting  is  assumed  to  come  entirely  from  the  hot  calorimeter 
and  water,  and  to  be  equal  to  the  heat  lost  by  them.  Form  the 
heat  equation  containing  all  these  quantities  of  heat,  and  solve 
it  for  Z,  which  is  used  to  denote  the  latent  heat  of  fusion  of  ice 
(or  the  latent  heat  of  water).  If  the  calorimeter  is  of  brass, 
take  .094  for  its  specific  heat;  if  of  other  material,  find  its 
specific  heat  in  Table  VIII  of  the  Appendix. 


LATENT   HEAT   OF    FUSION   OF   ICE 


115 


OBSERVATIONS 

a.  Weight  of  calorimeter 

b.  Weight  of  calorimeter  and  initial  water 

c.  Initial  temp,  of  calorimeter  and  water 
Final  temp,  of  calorimeter  and  water 

d.  Final  weight  of  calorimeter  and  water 

(including  water  from  ice) 

COMPUTATIONS 

Weight  of  water  before  adding  ice 

Weight  of  ice  added 

Fall  of  temperature  of  calorimeter  and 

initial  water 

Loss  of  heat  by  calorimeter 
Loss  of  heat  by  initial  water 
Heat  required  to  melt  the  ice  =  (  )  x  L 
Heat  required  to  raise  temperature  of 

melted  ice  from  0°  to  final  temp. 
Latent  heat  of  fusion  of  ice  (L) 
True  value  of  latent  heat  of  fusion  of  ice 
Per  cent  of  error 


= g- 


'C. 


=  — g. 


= g- 


— calories 

—  calories 
= calories 

—  calories 
—  calories 
80  calories 


EXEKCISE  39. 


LATENT    HEAT    OF    VAPORIZATION 
OF   WATER 


References.  —  Hoadley,  285-286;  Carhart  and  Chute,  342; 
Slate,  152. 

To  find  the  number  of  calories  given  out  by  1  g.  of 
steam  at  100°  in  condensing  to  water  at  100°. 

Apparatus.  —  Calorimeter ;  steam  generating  apparatus ;  Bun- 
sen  burner;  rubber  tube  and  condensation  trap  (Fig.  40)  or 
side-neck  test  tube ;  thermometer  and  stirrer ;  platform  balance 
and  weights ;  mop  cloth. 


116  HEAT 

a.    Fill  the  steam  generator  about  half  full  of  water,  and 
heat.     Connect  the  delivery  tube  and  condensation  trap.     Sup- 
port the  delivery  tube  on  some  object  so  that  the  escap- 
ing steam  will  not  damage  the  table. 

b.  Weigh  the  calorimeter  to  .1  g.     (It  is  especially 
important   in   this    experiment  that  the  weighing  be 
carefully  done.) 

c.  Fill  the  calorimeter  about  two  thirds  full  of  water 
at  5°  to  8°.     Add  ice  to  water  from  the  faucet  till  the 

.  40.  required  temperature  is  obtained.  (If  no  ice  is  pro- 
vided, use  the  coldest  water  obtainable ;  and  carry  the  tem- 
perature with  the  steam,  —  as  directed  in  the  next  paragraph, 
—  up  to  about  40°.)  Weigh  carefully  the  calorimeter  and  con- 
tents, and  remove  from  the  balance. 

d.  Place  the  stirrer  on  the  calorimeter ;  and,  as  soon  as  the 
stearn  is  escaping  freely  from  the  delivery  tube,  stir  the  water 
in  the  calorimeter  and  take  the  temperature,  and  immediately 
place  the  delivery  tube  in  it  to  a  depth  of  an  inch  or  two.     The 
calorimeter  should  be  as  far  as  possible  from  the  burner,  and 
protected  from  its  radiation  by  a  screen.     Stand   your  note- 
book between  them  for  this  purpose.     There  should  be  no  con- 
siderable loss  of  steam  on  account  of  poorly  adjusted  apparatus. 
If  the  steam  is  being  delivered  properly,  the  temperature  will 
rise  rapidly. 

Stir  the  water  continuously,  keeping  the  hands  off  the  calo- 
rimeter. Be  careful  not  to  let  the  condensation  trap  over- 
flow and  admit  hot  water  into  the  calorimeter.  To  empty  it, 
remove  the  burner  from  under  the  boiler  and  lift  the  delivery 
tube  till  the  water  in  the  trap  runs  back  into  -the  boiler. 

When  the  temperature  is  about  as  far  above  the  tem- 
perature of  the  room  as  it  was  below  it  at  the  start,  turn 
off  the  gas,  remove  the  delivery  tube,  and  immediately, 
while  stirring,  observe  the  highest  uniform  temperature 
attained. 

e.  Weigh  the  calorimeter  and  contents. 


LATENT    HEAT   OF   VAPORIZATION   OF   WATER       117 


Repeat  the  experiment  if  there  is  time.  Leave  the  calo- 
rimeter empty  and  the  table  dry. 

/.  How  has  loss  or  gain  of  heat  by  radiation  and  conduction 
been  provided  for  in  this  experiment  ?  L  is  used  to  denote  the 
latent  heat  of  vaporization  of  water ;  which  is  also  the  amount 
of  heat  given  out  by  1  g.  of  steam  in  condensing  to  water  at 
100°.  Write  the  heat  equation  and  solve  for  L,  as  in  the  pre- 
ceding experiment. 

OBSERVATIONS 


b.  Weight  of  calorimeter 

c.  Weight  of  calorimeter  and  initial  water     = 

d.  Initial  temp,  of  calorimeter  and  water 
Final  temp,  of  calorimeter  and  water  = 

e.  Final  weight  of   calorimeter  and   water 

(including  water  from  steam) 

COMPUTATIONS 

Weight  of  water  before  adding  steam 

Weight  of  steam  added 

Eise  of  temperature  of  calorimeter   and 

initial  water 

Gain  of  heafc  by  calorimeter 
Gain  of  heat  by  initial  water  = 

Heat  given  out  by  steam  in  condensing  to 

water  at  100°  =  (  )  X  L 
Heat  given  out  by  water  from  steam  in 

falling  to  final  temperature 
Latent  heat  of  vaporization  of  water  (L)       = 
True  value  of  latent  heat  of  vaporization 

of  water 
Per  cent  of  error 


—  calories 

—  calories 


-  calories 

-  calories 

-  calories 


=    r>37  calories 


VI.     SOUND 
EXEECISE  40.     THE   TRANSMISSION   OF   SOUND 

References.  —  Hoadley,  181-184;  Carhart  and  Chute,  174- 
179  and  198-199 ;  Jones,  Sound,  10. 

Apparatus  (for  Parts  I  and  II).  —  Meter  rod ;  tuning  fork  ] ; 
tumbler  or  jar  of  water ;  large  cork  or  small  block  of  wood 
with  hole  to  fit  the  stem  of  the  fork ;  rubber  mallet  for  strik- 
ing the  fork ;  an  apoustic  telephone. 

[For  a  rubber  mallet  bore  a  half  inch  hole  in  a  large  rubber  stopper 
and  insert  a  stick  about  10  in.  long  for  a  handle  ;  or  slip  a  short  piece  of 
large,  thick  rubber  tubing  on  the  end  of  a  stick.  To  make  an  acoustic 
telephone,  make  a  small  hole  in  the  middle  of  the  bottom  of  two  small 
tin  cans  or  chalk  boxes,  fasten  them  up  at  some  distance  apart,  and 
stretch  a  cord  or  small  wire  rather  tightly  between  them,  fastening  the 
ends  to  some  small  object  (a  button)  on  the  inside  of  the  bottom  of  the 
cans.  The  cord  must  not  be  supported  by  fastening  it  rigidly  to  any 
object.  It  may  be  supported  at  any  point  by  a  cord,  and  may  be  carried 
round  a  corner  by  giving  it  three  or  four  supports  at  the  corner,  making 
each  bend  slight.  It  is  rather  better  to  replace  the  bottom  of  the  can  or 
box  by  a  piece  of  parchment.] 

I.  To  investigate  the  transmission  of  sound  through 
solids. 

a.  To  set  a  tuning  fork  in  vibration,  hold  it  by  the  stem  in 
one  hand,  and  strike  one  of  the  prongs  a  sharp,  quick  blow 
near  its  end  in  the  direction  to  drive  it  toward  the  other 

1  Tuning  forks  for  experimental  purposes  should  be  large  and  heavy. 
Small  forks  do  not  vibrate  long  enough  nor  with  sufficient  energy. 

118 


THE   TRANSMISSION   OF   SOUND  119 

prong.  Hold  one  end  of  the  meter  rod  (or  other  long  stick) 
close  to  the  ear  while  your  companion  holds  the  stem  of  the 
vibrating  fork  against  the  other  end  of  the  rod.  Remove  the 
ear  from  the  rod  and  listen  for  the  sound  of  the  fork  through 
the  air. 

Compare  the  loudness  of  the  sound  transmitted  through  the 
rod  and  through  the  air. 

b.  Hold  the  end  of  the  rod  between  your  teeth  while  the 
fork  is  sounded  against  the  other  end.     The  vibrations  travel 
through  the   rod,  the   teeth,  and  the  bones  of  the  head  to 
the  ear. 

Describe  the  result.     Do  you  feel  the  vibrations  ? 

c.  Hold  the  stem  of  the  vibrating  fork  against  the  teeth ; 
against  the  top  of  the  head.     State  the  result. 

d.  Tie  a  string  one  or  two  meters  long  to  the  stem  of  the 
fork.     Press  one  end  of  the  string  into  your  ear  while  your 
companion  sets  the  fork  vibrating  and  holds  it  so  as  to  stretch 
the  string  moderately  tight.     Try  the  effect  of  slackening  the 
string  and  of  removing  it  from  the  ear. 

What  have  you  learned  about  the  transmission  of  sound  by 
the  string  ? 

e.  Touch  the  stem  of  the  vibrating  fork  to  the  table  top. 
The  loud  sound  comes  from  the  table,  which  is  set  in  vibration 
by  the  fork.     Hold  an  end  of  the  meter  rod  against  the  side 
of  the  table,  and  the  vibrating  fork  against  the  other  end  of 
the  rod. 

State  and  account  for  the  result. 

/.  Place  the  rubber  stopper  of  the  mallet  between  the  vibrat- 
ing fork  and  the  table. 

How  does  the  stopper  compare  with  the  rod  in  its  power  to 
transmit  sound  ?  Account  for  the  difference. 

g.  If  an  acoustic  telephone  is  set  up  in  the  laboratory,  listen 
at  one  end  of  it  while  your  companion  sounds  the  fork  against 
the  bottom  of  the  can  or  box  at  the  other  end.  Try  speaking 
to  each  other  through  the  telephone. 


120  SOUND 

II.  To  find  whether  water  transmits  sound. 

Place  a  tumbler  of  water  on  the  table.  Insert  the  stem  of 
the  fork  into  the  hole  in  the  cork  (or  block).  Set  the  fork  in 
vibration  and  hold  it  with  the  cork  immersed  in  the  water, 
but  not  touching  the  glass.  Raise  the  cork  out  of  the  water 
and  again  immerse  it,  repeating  the  process  a  number  of  times, 
and  note  the  effect  on  the  loudness  of  the  sound.  State  the 
result. 

With  the  fork  sounding  and  its  stem  in  the  water,  try  the 
effect  of  lifting  the  tumbler  from  the  table  and  again  replacing 
it.  State  and  account  for  the  effect  upon  the  sound  when  the 
tumbler  is  lifted. 

How  does  the  experiment  answer  the  question  whether 
water  transmits  sound? 

III.  To  investigate  the  transmission  of  sound  through 
a  speaking  tube. 

Apparatus.  —  A  tin  or  large  glass  tube  6  ft.  to  10  ft.  long ;  a 
roll  of  cotton  or  soft  cloth. 

Lay  a  watch  on  a  roll  of  cotton  or  soft  cloth  (to  prevent 
transmission  of  the  sound  through  the  table)  near  one  end  of 
the  tube,  and  listen  at  the  other  end  for  the  sound  of  the 
ticking. 

About  how  near  to  the  watch  must  you  hold  the  ear  to  hear 
it  as  distinctly  directly  through  the  air  as  through  the  tube  ? 

Explain  the  effect  of  the  tube. 

EXERCISE  41.     THE  VELOCITY   OF   SOUND   IN  AIR 

References.  —  Hoadley,  189-192;  Carhart  and  Chute,  180- 
183 ;  Jones,  Sound,  24-27. 

Apparatus.  —  Pendulum  that  beats  half  seconds  ;  hammer  ; 
piece  of  iron  or  other  object  that  gives  a  loud,  sharp  sound 
when  struck ;  tape  measure  or  long  cord  of  measured  length. 


THE    VELOCITY   OF    SOUND   IN   AIR  121 

To  determine  the  velocity  of  sound  in  open  air  by 
timing  an  echo. 

a.  This  method  is  to  be  preferred  if  there  is  a  large  building 
that  can  be  utilized  as  a  reflector  for  obtaining  an  echo.     This 
will  require  a  free  space  of  about  300  ft.  beside  the  building. 

Adjust  a  pendulum  to  beat  half  seconds.  It  will  be  about 
25  cm.  long ;  but  find  the  exact  length  by  trial.  A  stone  tied 
to  a  string  will  serve. 

Stand  facing  the  building  at  a  considerable  distance  from  it 
and  in  line  with  the  perpendicular  to  the  middle  of  the  side  of 
the  building.  Start  the  pendulum  swinging.  Hold  the  hammer 
in  one  hand  and  the  object  to  be  struck  in  the  other ;  swing  the 
hammer  to  and  fro  horizontally  in  exact  time  with  the  pendu- 
lum ;  and  strike  the  object  once  per  second  exactly  at  the  instant 
that  the  pendulum  completes  its  swing  to  one  side  (the  left). 

b.  Listen  for  the  echo,  and  find  by  trial  the  distance  from 
the  building  at  which  you  hear  it  exactly  at  the  instant  the 
pendulum  completes  its  swing  to  the  opposite  side  (the  right). 
The  echo  must  divide  the  time  between  the  blows  exactly  in 
half.     Repeat  the  blows  many  times  in  succession,  carefully 
watching  the  pendulum  and  listening  to  the  echo. 

c.  When  you  have  found  the  distance  from  the  building  that 
gives  the  best  results,  measure  it.     Since  the  echo  was  heard 
one  half  second  after  the  blow  was  struck,  the  sound  traveled 
twice  this  distance  (to  the  building  and  back)  in  one  half  a 
second.     Hence  four  times  this  distance  is  your  experimental 
value  of  the  velocity  of  sound  at  the  temperature  when  the 
experiment  was  performed. 

d.  Record  the  temperature  if  you  have  access  to  a  thermom- 
eter ;  if  not,  record  your  estimate  of  the  temperature. 

e.  Compute  the  true  value  of  the  velocity  of  sound  from  the 
formula  v  =  (1090  +  p°  _  32°]  )  ft., 

in  which  t  is  the  temperature  by  the  Fahrenheit  thermometer ; 
and  from  this  find  the  per  cent  of  error  of  your  result. 


122  SOUND 

To  determine  the  velocity  of  sound  in  open  air  by 
timing  a  sound  made  at  a  distance. 

a.  This  method  requires  two  stations  between  500  and  600  ft. 
apart  with  an  unobstructed  view  between   them.     It   is  de- 
sirable to  have  an  opera  glass  or  a  spy  glass  for  observing 
the  pendulum  at  a  distance. 

At  one  station  set  up  a  pendulum  that  beats  half  seconds 
(see  paragraph  a  of  the  first  method)  with  a  screen  behind 
it  so  that  it  can  be  more  easily  seen  at  a  distance.  Use  a 
white  screen  if  the  bob  of  the  pendulum  is  dark  and  a  black 
screen  if  it  is  light.  The  bob  must  be  large  if  it  is  to  be 
observed  with  the  naked  eye. 

One  person  stands  beside  the  pendulum  and  makes  a  loud 
sound  once  per  second  as  directed  in  paragraph  a  of  the  first 
method  ;  and  another  finds  by  trial  the  distance  at  which  he 
hears  the  sound  exactly  one  half  second  after  the  blow  is  struck; 
that  is,  at  the  instant  the  pendulum  completes  its  swing  to  the 
opposite  side.  As  noted  above,  it  will  be  better  to  use  an 
opera  glass  or  a  spy  glass  for  observing  the  pendulum. 

b.  Exchange  places  and  repeat  the  experiment.     It  will  be 
evident  that  the  probable  error  in   determining  the  distance 
is  rather  large  ;  but  the  estimates  of  the  two  observers  should 
agree  within  50  ft.     By  repeated  trial  secure  agreement  within 
this  limit  if  possible. 

c.  Measure  the  distance    between  the  stations,  taking  the 
average  if  the  estimates  differ;  and  double  this  distance  for 
the  velocity  of  sound  at  the  temperature  when  the  experiment 
was  performed. 

d.  Kecord  the  temperature  if  you  have  access  to  a  thermom- 
eter ;  if  not,  record  your  estimate  of  the  temperature. 

e.  Compute  the  true  value  of  the  velocity  of  sound  from  the 
formula 


o  _  32°-]  )  f  t>  ^ 

in  which  t  is  the  temperature  by  the  Fahrenheit  thermometer  ; 
and  from  this  find  the  per  cent  of  error  of  your  result, 


THE    REFLECTION   OF    SOUND  123 

EXERCISE   42.     THE   REFLECTION   OF   SOUND1 

i 

References.  —  Hoadley,  193-195 ;  Carhart  and  Chute,  184- 
186. 

I.  To  study  the  reflection  of  sound  from  a  plane  sur- 
face. 

Apparatus.  —  Two  large  glass  or  tin  tubes,  2  to  3  ft.  long ; 
screen  or  other  vertical  plane  surface  to  serve  as  a  reflector; 
roll  of  cotton  or  soft  cloth ;  watch. 

a.  Lay  the  tubes  on  the  table  so  as  to  form  an  angle  between 
50°  and  60°,  the  ends  at  the  vertex  of  the  angle  being  close 
together  but  not  touching.     Lay  a  watch 

on  the  roll  of  cotton  at  the  end  A  (Fig.  41), 
of  one  of  the  tubes,  or  just  inside  it  if 
the  tube  is  large  enough.  Be  sure  that  the 
watch  does  not  touch  the  tube.  Listen  at 
B  for  the  ticking,  first  with  the  reflector 
in  position  at  (7,  making  equal  angles 
with  the  tubes,  then  with  the  reflector 
removed. 

State  and  account  for  the  result  in  each  case. 

b.  While  listening  at  B  as  before,  gradually  turn  the  re- 
flector about  a  vertical  axis  till  it  makes  very  unequal  angles 
with  the  two  tubes.     State  the  result. 

c.  Set  the  tubes  at  a  wider  angle  (80°  to  90°)  and  repeat 
paragraphs  a  and  b. 

d.  What  do  you  learn  from  the  experiment  concerning  the 
direction  in  which  sound  is  reflected  from  a  plane  surface? 

II.  To  study  the  reflection  of  sound  from  concave  sur- 
faces. 

Apparatus.  —  Two  concave  reflectors;  large  funnel  with  rub- 
ber tube  attached,  for  use  as  an  ear  trumpet ;  watch. 

1  This  exercise  can  be  performed  only  in  a  very  quiet  room.  It  should 
be  set  up  in  a  room  by  itself,  if  possible. 


124  SOUND 

a.  Stand  one  of  the  reflectors  (yi/Fig.  42)  at  one  end  of  the 
table,  and  turn  it  so  as  to  face  toward  the  other  reflector  (.B), 
placed  at  a  distance  of  3  or  4  m.  B  is  set  obliquely,  as  shown 
in  the  figure.  Hang  a  watch  in  front  of  the  center  of  reflector 
A  Sit  a,  distance  from  it  equal  to  about  half  the  radius  of  the 
spherical  surface.  This  point  (F)  is  called  the  focus  of  the 


FIG.  42. 

reflector.  It  is  easily  found  by  turning  the  reflector  toward 
the  sun  and  catching  the  reflected  light  on  a  piece  of  paper. 
By  moving  the  paper  to  and  fro,  find  the  position  where  the 
spot  of  light  is  the  smallest.  This  is  the  focus. 

Hold  the  ear  at  jEJ,  being  careful  to  cover  as  little  of  the 
reflector  with  the  head  as  possible.  Move  the  head  slightly 
in  different  directions  to  find  the  position  where  the  sound 
is  loudest.  When  the  ear  is  properly  placed  the  watch  should 
be  heard  distinctly. 

Instead  of  placing  the  ear  at  E,  the  reflector  B  may  be 
turned  so  as  to  face  squarely  toward  A,  and  the  ear  trumpet 
used  to  convey  the  sound  to  the  ear.  Place  the  funnel  at  the 
focus  of  B  and  facing  toward  it,  arid  the  end  of  the  tube  in 
the  ear.  Try  both  ways. 

b.  With  the  ear  at  E  or  with  the  ear  trumpet  in  position, 
observe  the  effect  on  the  loudness  of  the   sound  when   your 
companion   moves  the  watch  closer  to  and    farther  from    A. 
State  the  result  in  each  case. 

c.  With  the  ear  in  position  as  before,  observe  the  effect  of 
turning  A  about  a  vertical  axis  toward  one  side  and  toward 
the  other. 

In  what  direction  does  A  reflect  the  sound  most  distinctly  ? 


SYMPATHETIC    AND   FORCED   VIBRATIONS 


125 


EXERCISE   43. 


SYMPATHETIC   AND   FORCED 
VIBRATIONS 


References.  —  Hoadley,  198-199  and  202-203;  Carhart  and 
Chute,  189-192 ;  Slate,  185 ;  Sanford,  p.  211,  on  Forced  Vibra- 
tion; Jones,  Sound,  56-57. 

I.  To  study  sympathetic  and  forced  vibrations  (not 
sonorous}  by  means  of  pendulums. 

Apparatus.  —  Four  pendulums  supported  as  shown  in  Fig.  43. 
CD  is  a  light  rod  2  or  3  ft.  long,  suspended  by  short  cords  from 
any  convenient  fixed  sup-  fe^.^£^s^as_g=^..^^.J^rf==,_.3ja,, ,  „. 


_L 


T 


FIG.  43. 


port.  From  CD  two  pairs 
of  pendulums  are  sus- 
pended, the  pendulums  of 
each  pair  being  of  equal 
length  and  the  shorter  pair 
about  half  the  length  of 
the  longer. 

a.  Set  one  of  the  longer 
pendulums  vibrating  in  the 
direction  of  the  supporting 
rod    (CD,    Fig.    43),    and 

observe  the  effect  upon  the  other  pendulums.  Describe  and 
account  for  the  observed  effects,  noting  particularly  any 
difference  in  the  effect  upon  the  longer  and  the  shorter 
pendulums. 

What  examples  of  sympathetic  or  of  forced  vibrations  are 
afforded  by  the  motions  of  the  pendulums  ? 

b.  Bring  the  pendulums  to  rest  and  repeat,  this  time  start- 
ing one  of  the  shorter  pendulums.     Describe  and  discuss  the 
result. 

c.  Again  bring  the  pendulums  to  rest  and  set  a  long  one  and 
a  short  one  vibrating  at  the  same  time.     Describe  and  account 
for  the  motion  of  the  other  two  pendulums. 


120  SOUND 

d.  Bring  the  pendulums  to  rest  and  give  the  supporting  rod 
a  slight  push  in  the  direction  of  its  length.  Observe  its  rate 
of  vibration. 

Is  it  the  same  as  that  of  any  of  the  pendulums  ? 

Are  the  motions  impressed  upon  it  by  the  pendulums,  when 
vibrating,  examples  of  forced  or  sympathetic  vibrations  ? 

II.  To  study  the  sympathetic  vibration  of  tuning  forks 
and  resonators. 

Apparatus.  —  Two  tuning  forks  of  exactly  the  same  pitch 
(shown  by  the  absence  of  beats  when  sounded  together) ; 
rubber  mallet;  soft  wax;  short  pieces  of  large  glass  tubing 
of  different  length  and  diameter. 

[To  make  the  soft  wax,  melt  together  about  nine  parts,  by  weight,  of 
beeswax  and  one  part  of  Venice  turpentine.  Forks  giving  a  few  beats 
per  second  may  be  permanently  tuned  to  unison  by  filing  a  little  off  the 
inside  of  the  prongs  at  the  base  of  the  higher  fork  or  the  free  ends  of  the 
lower  one.  It  will  be  more  interesting  if  the  glass  tubes  are  of  such  sizes 
as  to  sound  a  major  chord.  The  longer  tubes  should  have  the  greater 
diameter.] 

a.  Sound  one  of  the  forks  and  hold  it  and  the  other  fork 
close  together,  facing  each  other,  but  not  touching.     After  one 
or  two  seconds  hold  the  fork  that  was  silent  close  to  the  ear. 
It  will  be  found  to  be  vibrating  audibly.     Explain. 

b.  Sound  one  of  the  forks  and  hold  the  stems  of  both  against 
the  table  top.     After  one  or  two  seconds  stop  the  fork  that  was 
sounded.     The  other   fork    should  now  be  sounding  audibly. 
If  it  is  not,  try  again  until  you  are  successful.     How  was 
the  vibration  set  up  in  the  second  fork  ? 

c.  Stick  a  bit  of  wax  about  twice  the  size  of  a  pea  near  the 
end  of  a  prong  of  one  of  the  forks.     This  will  slightly  change 
the  rate  of  vibration  of  the  fork,  as  can  be  shown  by  sounding 
both  of  the  forks  and  holding  them  side  by  side  near  the  oar. 
The  pulsation  of  the  sound  (called  beats)  is  due  to  a  slight 
difference  in  the  vibration  rates  of  the  forks.     Now  repeat  the 


SYMPATHETIC   AND   FORCED    VIBRATIONS  1:27 

experiments  of  paragraphs  a  and  b,  —  sounding  either  of  the 
forks,  — and  observe  whether  the  silent  fork- is  made  to 
vibrate.  State  and  explain  the  result. 

d.  Blow  across  the  ends  of  the  glass  tubes  in  succession. 
Observe  that  each  tube  gives  forth  a  sound  of  definite  pitch. 
Hold  the  tubes,  two  at  a  time,  close  to  the  ears,  and  note  the 
faint  sounds,  like  the  roar  of  a  sea  shell,  coming  from 
them. 

How  does  the  pitch  of  the  sound  coming  from  each  tube 
compare  with  that  produced  by  blowing  across  the  end  of  it  ? 

Account  for  these  faint  sounds  coming  from  the  tubes. 

III.  To  study  the  sympathetic  and  forced  vibration 
of  a  sonometer  wire. 

Apparatus.  —  Sonometer  with  two  wires  of  the  same  size  and 
without  a  bridge. 

a.  Tighten  one  of  the  sonometer  wires  to  a  moderate  ten- 
sion, and  tune  the  other  wire  in  perfect  unison  with  it.     When 
the  wires  are  nearly  in  unison,  listen  for  a  periodic  pulsation 
of  the  sound  (beats)  when  both  wires  are  plucked.     As  the 
sounds  approach  unison  the  pulsations   become   slower,    and 
they  disappear  when  unison  is  secured.     Tune  the  wires  till 
the  pulsations  cease.     Now  sound  one  of  the  wires  and  the 
other  will   immediately  begin   to  vibrate  visibly.      Stop  the 
first  wire  and  the  sound  will  be  continued  with  considerable 
intensity  by  the  other. 

Is  this  a  case  of  sympathetic  or  forced  vibration?  How 
was  it  set  up? 

b.  With  the  wires  of  the  sonometer  so  nearly  in  unison  that 
they  give  less  than  one  pulsation  per  second  when  sounded 
together,  sound  only  one  of  them  and  observe  the  behavior 
of  the  other.     It  should  vibrate  visibly  for  brief  intervals, 
which  alternate  with  intervals  of   rest.     Observe   that  these 
alternations  of  rest  and  vibration  become  more  rapid  as  the 
difference  in  pitch  is  increased;  the  amplitude  simultaneously 


128  SOUND 

decreasing,  until  presently  the  wire-  ceases  to  respond  at  all 
to  the  vibrations  of  the  other. 

Account  for  the  behavior  of  the  second  wire  under  the  dif- 
ferent conditions,  and  compare  with  the  experiments  with  the 
pendulums. 

EXERCISE  44.     WAVE   LENGTH   BY  RESONANCE 

References.  — Hoadley,  198-201;  Carhart  and  Chute,  189- 
194 ;  Jones,  Sound,  62. 

Apparatus.  —  Some  form  of  resonance  tube  or  jar  with 
adjustable  length  (Figs.  44-47);  one  or  more  tuning  forks; 
rubber  mallet ;  rubber  band ;  meter  rod. 

I.  To  find,  by  ^neans  of  a  resonance  tube,  the  length 
of  tlw  sound  waves  set  up  by  a  tuning  fork  of  known 
vibration  rate. 

Theory.  —  From  a  study  of  the  text  you  will  learn  that  the 
resonance  tube  sounds  when  the  length  of  the  confined  air 
column  is  very  nearly  equal  to  one  fourth  the  length  of  the 
waves  set  up  by  the  fork  used ;  and  that  the  "  correction  for 
the  diameter"  is  one  half  the  diameter  or  the  radius  of  the 
tube,  which  is  to  be  added  to  the  length  of  the  air  column. 
The  tube  will  again  sound  when  the  length  is  further  increased 
by  exactly  half  a  wave  length.  Let  L  denote  the  wave  length 
of  the  fork,  l:  the  length  of  the  air  column  for  first  resonance, 
and  ?2  f°r  second  resonance,  and  r  the  radius  of  the  tube; 

then  L  =  4  (^  +  r),  for  first  resonance, 

and  L  =  2  (12  —  7^,  for  second  resonance. 

The  latter  will  probably  be  more  exact,  as  it  does  not  involve 
the  correction  for  the  diameter  of  the  tube,  which  is  somewhat 
uncertain. 


WAVE   LENGTH   BY    RESONANCE  1^9 

DIRECTIONS  FOR  TUBE  WITH  PISTON 

a.  It  is  better  for  two  students  to  work  together  in  this 
exercise,  one  operating  the  piston  and  the  other  the  fork. 
Sound  the  fork  and  hold  it  at  the  end  of  the  tube  in  the  posi- 
tion shown  in  Fig.  44.  At  the  same  time,  starting  with  the 
piston  near  the  same  end  of  the  tube,  pull  it  slowly  and  steadily 
back  till  the  tube  responds  to  the  vibrating  fork.  Mark  this 
position  of  the  front  of  the  piston  with  the  rubber  band  on 
the  tube.  Move  the  piston  back  and  forth  several  times  past  the 
position  of  maximum  reinforcement,  gradually  diminishing  the 


FIG.  44. 

range  of  motion.  When  you  have  secured  the  best  adjustment 
possible,  measure  the  distance  from  the  end  of  the  tube  to  the 
piston.  This  is  recorded  as  the  length  of  the  air  column  (IJ. 
Move  the  rubber  band  and  piston  out  of  position,  and  make 
a  second  and  entirely  independent  trial.  If  it  does  not  differ 
from  the  first  by  more  than  3  mm.,  the  work  is  sufficiently 
accurate,  and  the  average  of  the  two  results  may  be  taken  as 
the  correct  value.  If  the  difference  is  more  than  this,  repeat 
till  consistent  results  are  obtained. 

b.  In  the  same  way  find  the  position  of  the  piston  for  second 
resonance.     The  correct  position  can  be  more  quickly  found  by 
remembering  that  the  air  column  will  now  be  approximately 
three  times  as  long  as  before.     Make  at  least  two  trials,  and 
more  if  necessary,  as  before. 

c.  Measure  the  diameter  of  the  tube  and  take  the  temperature 
of  the  room.    Kecord  the  vibration  number  (2V)  of  the  fork  used. 
Compute  the  wave  length  by  both  formulas  as  indicated  below. 

COLEMAN'S  PHY.  LAB.  MAN.  —  9 


180 


SOUND 


cm. 

•  cm. 

cm. 


FORM  OF  RECORD 

a.  Length  of  air  column  for  first  resonance  = 
Ditto,  second  trial  = 
Ditto,  average  value  (7j) 

b.  Length  of  air  column  for  second  resonance  = cm. 

Ditto,  second  trial  =  —   —  cm. 

Ditto,  average  value  (72)  = cm. 

c.  Radius  of  the  tube  = 

Temperature  of  the  room  = °  C. 

Vibration  number  of  the  fork  (N)  = 

Length  of  wave  (L)  =  4  (^  -f  r)  = cm. 

Length  of  wave  (L)  =  2  (12  —  li)  = cm. 

DIRECTIONS  FOR  APPARATUS  SHOWN  IN  FIG.  45 

a.  With  the  third  clamp,  support  the  funnel  so  that  its  top 
is  about  lo  cm.  below  the  top  of  the  glass 
tube.  Pour  water  into  the  funnel  till  it 
stands  about  half  full.  Remove  the  funnel 
from  the  clamp ;  hold  it  in  the  hand ;  and 
while  the  vibrating  fork  is  held  just  above 
the  tube,  raise  and  lower  the  funnel,, causing 
the  water  to  rise  and  fall  in  the  tube  till  the 
adjustment  giving  the  loudest  reenforcement 
of  the  sound  is  obtained.  Mark  the  height 
of  the  water  in  the  tube  by  the  rubber  band. 
Cause  the  water  to  rise  and  fall  several  times 
past  the  position  of  greatest  reenforcement, 
each  time  trying  to  adjust  the  position  of 
the  rubber  band  more  accurately.  Measure 
the  length  of  the  air  column  and  record  the 
result  as  the  first  trial. 

Repeat  the  process  of  adjusting  the  rubber 
band,  first  displacing  it  several  centimeters, 
FIG.  45.  so  that  the  judgment  will  not  be  influenced 


WAVE   LENGTH   BY   RESONANCE 


181 


by  the  first  trial.  If  the  second  result  does  not  differ  from 
the  first  by  more  than  3  inm.,  the  work  is  sufficiently  accurate, 
and  the  average  of  the  two  results  may  be  taken  as  the  cor- 
rect value.  If  the  difference  is  more  than  this,  repeat  till 
consistent  results  are  obtained. 

b.  Draw  off  the  water  in  the  tube  till  the  funnel  is  about 
half  full  when  the  air  column  is  about  three  times  as  long  as 
for  first  resonance.     Find  the  length  of  the  air  column  for  sec- 
ond resonance  in  the  same  manner  as  before,  repeating  till  two 
results  are  obtained  which  do  not  differ  by  more  than  3  mm. 
When  you  have  finished,  clamp  the  funnel  in  place. 

c.  Follow  the  directions  of  paragraph  c  above  and  the  above 
form  of  record. 


DIRECTIONS  FOB  OTHER  FORMS  OF  EESONATORS 

a.  Insert  a  glass  cylinder  (student  lamp  chimney)  into  a  bat- 
tery or  hydrometer  jar  filled  with  water  (Fig.  46).     The  length 

of   the   air   column  is  varied 

by  raising   and  lowering   the 

cylinder. 

Or  an  hydrometer  jar  may 

be  used  for  the  resonator,  and 

the  level  of   the  water  in  it 

adjusted  by  means  of  a  siphon, 

using  a  rubber  tube  for  this 

purpose  (Fig.  47). 

In    either    case    read   para- 
graph a  of  the  first  directions 
for  the  general  plan  of  the  experiment,  and  make  the  obvious 
modifications  of  the  method  of  procedure. 

b.  Both  the  tube  and  the  jar  are  too  short  for  second  reso- 
nance.   Instead  of  this  part,  the  wave  length  of  a  fork  of  different 
pitch  may  be  found,  using  first  resonance  as  before. 

c.  Follow  the  directions  of  paragraph  c  of  the  first  directions. 


FIG.  46. 


FIG.  47. 


132  SOUND 

II.  To  compute  the  velocity  of  sound  from  the  data  of 
Part  I. 

a.  From  the  known  vibration  number  (N)  of  the  fork,  and 
the  value  of  L  determined  by  the  experiment,  compute  the 
velocity  of  sound  at  the  temperature  of  the  room  from  the 
relation  v  =  NL. 

b.  Compute  the  true  value  of  v  at  the  temperature  of  the 
room  from  the  formula  v  =  (332  -f-  .6 1°)  meters,  in  which  t  is 
the  temperature  of  the  room  by  the  Centigrade  thermometer. 

c.  Compute  the  per  cent  of  error  of  your  result. 

EXERCISE  45.     INTERFERENCE   AND   BEATS 

References.  — Hoadley,  196-197  and  204 ;  Carhart  and  Chute, 
201-203 ;  Sanf  ord,  p.  215,  Interference  of  Sound  Waves  to  (6). 

Apparatus.  —  Two  tuning  forks  giving  from  one  to  three  beats 
per  second ;  paper  cylinder  about  10  cm.  in  length  and  2  cm.  in 
diameter ;  rubber  mallet ;  soft  wax. 

I.  To  study  the  interference  of  the  sound  waves  about 
a  tuning  fork. 

a.  Hold  a  vibrating  fork  near  the  ear  and  parallel  to  the 
face,  and  rotate  it  slowly.     Your  companion  will  tell  you  the 
position  of  the  fork  when  the  sound  is  loudest  and  when  it  is 
faintest  to  you.     How  many  times  does  the  sound  swell  and 
die  away  during  one  rotation  ? 

Can  you  find  positions  in  which  the  sound  is  inaudible  ? 

b.  With  the  vibrating  fork  held  to  the  ear  in  the  position  in 
which  the  sound  is  faintest,  let  your  companion  cover  one  of 
the  prongs  with  the  paper  cylinder,  being  careful  not  to  touch 
the  fork.     Repeat  till  you  are  sure  of  the  effect.     State  it. 

c.  Figure  48  represents  the  sound  waves  about  a  vibrating 
fork,  as  seen  with  the  ends  of  the  fork  pointing  toward  the 
observer.     The   prongs  always  move  toward  and  from  each 


INTERFERENCE   AND   BEATS 


133 


other  simultaneously  ;  hence,  in  separating,  a  condensed  half 
wave  is  set  up  on  the  outside  of  each,  and  a  rarefied  half  wave 
between  them  ;  and,  on  approaching  each  other,  the  opposite 
conditions  are  produced.  If  the  space  about  the  fork  were 
partitioned  off  into  four  compartments,  as  shown  at  the  left, 
there  would  be  condensations  and  rarefactions  on  opposite  sides 


FIG.  48. 


of  the  partitions,  at  equal  distances  from  the  fork.  But  as 
there  are  no  partitions  to  keep  the  condensations  and  rarefac- 
tions apart,  their  opposing  tendencies  are  mutually  destructive 
in  these  regions,  causing  silence,  as  shown  at  the  right. 

Why  was  sound  restored  when  one  of  the  prongs  was  cov- 
ered by  the  paper  cylinder  ? 

II.    To  study  beats  by  means  of  two  tuning  forks  of 
very  nearly  the  same  pitch. 

a.  Sound  both  forks  and  hold  them  facing  each  other  close 
to  the  ear,  in  the  position  for  loudest  sound.     Estimate  roughly 
the  frequency  of  the  beats.     Sound  the  forks  and  touch  them 
to  the  table.     Can  you  distinguish  the  beats  ? 

b.  Stick  a  small  bit  of  soft  wax  to  a  prong  of  one  of  the 
forks,  near  the  end,  and  observe  the  effect  on  the  frequency  of 
the  beats.     If  the  effect  is  too  small  to  be  noticed,  use  more 
wax.     It  is  better  to  stick  some  wax  on  both  prongs  than  a 


134 


SOUND 


large  quantity  on  one.  The  effect  of  the  wax  on  the  frequency 
of  the  beats  will  depend  upon  whether  it  has  been  put  on  the 
fork  of  the  lower  or  the  higher  pitch.  Prove  this  by  observ- 
ing the  beats  with  the  wax  on  the  other  fork. 

c.  Tune  the  forks  accurately  to  the  same  pitch  by  loading 
one  of  them  till  the  beats  cease. 

Does  loading  a  fork  raise  or  lower  its  pitch?      Why  ? 


FIG.  49. 

d.  Figure  49  presents  an  explanation  of  beats.  Copy  the 
top  and  middle  parts  of  it,  and  write  a  brief  explanation. 

Why  are  beats  less  frequent  as  unison  becomes  more  nearly 
perfect  ? 

EXERCISE   46.     VIBRATING   STRINGS. 
EFFECT   OF   LENGTH 

References.  —  Hoadley,  220-222;  Carhart  and  Chute,  210- 
212. 

To  find  the  relation  between  the  length  of  a  vibrat- 
ing string  and  its  pitch. 


VIBRATING   STRINGS.     EFFECT   OF   LENGTH  185 


Apparatus.  —  Sonometer;  rubber  mallet;  several  tuning 
forks,  including  c'  (256  vibrations),  and  c"  (512  vibrations); 
meter  rod. 

a.  Tighten  the  wire  till  a  length  of  60  cm.  or  more  vibrates 
in  unison  with  the  c'  fork.     To  make  the  sound  of  the  fork 
audible,  touch  its  stem  to  the  sonometer  or  to  the  top  of  the 
table.     The  wire  gives  a  better  sound  and  one  easier  to  com- 
pare with  the  fork  if  it  is  plucked  near  the  middle  with  the  end 
of  the  ringer  or  the  thumb  (not  the  nail).     Tune  the  wire  by 
varying  its  length  by  means  of  the  bridge.      When  the  wire 
and  fork  are  nearly  in  unison,  listen  for  beats  and  tune  till 
they  disappear.     Measure  the  length  of  the  wire ;  then  dis- 
place the  bridge,  tune  again  (without  changing  the  tension), 
and  again  measure.     (If  the  difference  is  more  than  3  mm., 
make  further  trials.)     The  tension  of  the  wire  must  remain  the 
same  throughout  the  experiment. 

b.  Adjust  the  length  of  the  wire  so  as  to  bring  it  successively 
into  unison  with  the  other  forks,  making  at  least  two  trials  for 
each.     Eecord  as  indicated. 

c.  By  length  ratio  for  any  tone  is  meant  the  ratio  of  the 
length  of  the  wire  for  that  tone  to   the  length   of  the  wire 
for  c'.     Compute  the  length  ratio  in  each  case. 

d.  From  the  law  of  lengths  and  the  known  vibration  ratios 
of  the  forks  used  (see  the  text),  find  the  true  values  of  the 
length  ratios. 

FORM  OF  RECORD 


TONI 

LENGTH 
OF  WIRE 

MEAN 
LENGTH 

LENGTH  RATIO 

KKKOI; 

PER  CENT 
OF  ERROR 

By  Exp. 

True 

c1 

r1 

e' 



rf 







.8 





etc. 

VII.     LIGHT 

EXERCISE   47.      SOME    RESULTS    OF    RECTILINEAR 
PROPAGATION 

References.  —  Hoadley,  443-445  and  447-448 ;  Carhart  and 
Chute,  231-236 ;  Jones,  Light,  4-8. 

Apparatus.  — A  flat  gas  jet  or  lamp  with  flat  wick  ;  optical 
bench  or  meter  rod  ;  a  screen  5  cm.  square,  mounted  on  a  wire 
(A,  Fig.  50)  ;  a  screen  (B)  with  horizontal  rows  of  holes,  and 
a  screen  (0)  with  a,  hole  in  the  center,  both  about  18  or  20  cm. 
square ;  short  metric  rule. 

[A  very  convenient  burner  for  this  exercise  and  the  one  on  photom- 
etry is  made  by  attaching  the  short  tube  and  tip  of  an  ordinary 
gas  jet  to  the  base  of  a  Bunsen  burner.  The  connection  is  made  air- 
tight with  a  little  paint  or  melted  wax.  It  will  be  most  convenient  to 
have  the  screens,  lenses,  spherical  mirrors,  diffusion  photometer  and 
candles  for  the  experiments  in  light  mounted  at  the  center  of  blocks 
of  uniform  size  in  which  a  groove  is  cut  so  that  they  will  fit  loosely 
upon  a  meter  rod  (placed  on  edge  or  lying  flat).  The  rod  thus  serves 
as  a  guide  to  keep  the  several  parts  of  the  apparatus  in  alignment, 
and  the  distances  between  them  will  be  the  distances  between  the  cor- 
responding ends  (right  or  left)  of  the  blocks  upon  which  they  are 
mounted.  A  board  110  cm.  long  and  8  or  10  cm.  wide,  with  the 
meter  rod  fastened  to  it,  makes  a  better  support  for  the  apparatus. 
Such  a  board  with  the  attached  rod  is  called  an  optical  bench  or, 
simply,  bench,  in  the  following  exercises.] 

I.    To  study  the  formation  of  shadows. 

a.  The  room  must  be  at  least  partially  darkened  for  this 
exercise.  Place  the  gas  jet  at  an  end  of  the  optical  bench  (or 

136 


SOME   RESULTS   OE   RECTILINEAR   PROPAGATION     137 


meter  rod),  and  place  the  screens  A  and  B  on  it  about  30  cm. 
and  80  cm.  respectively  from  this  end.  Turn  the  flame  first 
edgewise  then  flatwise  to  the  screens,  and  observe  the  appear- 
ance of  the  shadow  of  A  upon  B.  In  which  case  is  the  darker 
part  of  the  shadow  (the  umbra)  bordered  on  its  vertical  sides 
by  a  strip  of  fainter  shadow  (the  penumbra)  ? 

b.  Again  place  the  flame  flatwise,  and  adjust  the  distance  of 
B  so  that  the  line  between  the  umbra  and  the  penumbra  falls 
upon  B  at  the  break  in  the  line  of  holes.  The  upper  holes 
now  lie  in  the  penumbra,  and  the  lower  in  the  umbra  (Fig.  50). 
Look  toward  the  flame  through  each  of  the  holes  in  succes- 
sion. 

State  what  you  see,  —  (1)  when  looking  through  the  different 
upper  holes  ;  (2)  when  looking  through  the  lower  holes. 


FIG. 


From  your  observations  state  the  cause  of  the  umbra  and  of 
the  penumbra. 

Draw  figures  of  horizontal  sections  explaining  the  appear- 
ance of  the  shadows  with  the  flame  in  the  two  positions.  (Fol- 
low models  found  in  the  text  and  reference  books.  Drawings 
that  explain  are  not  mere  pictures  of  what  is  seen.) 

c.  Remove  A  and  in  its  place  hold  a  lead  pencil  vertically. 
Observe  the  shadow  of  the  pencil  with  the  flame  turned  edge- 
wise then  flatwise.  With  the  flame  in  the  latter  position, 
observe  the  varying  width  of  the  umbra  and  penumbra  as  the 


138  LIGHT 

screen  is  moved  toward  and  from  the  pencil.    Move  the  screen 
so  that  the  umbra  disappears. 

Describe  the  observed  changes  in  the  appearance  of  the 
shadow,  and  explain  them  with  the  aid  of  drawings. 

II.  To    study    the    formation    of    images    by    small 
openings. 

a.  Stand  the  gas  jet,  turned  flatwise,  at  the  end  of  the  optical 
bench.     Place  the- screens  B  and  C  on  the  bench  with  C  nearer 
the  light.     Move  the  screens,  together  and  separately,  toward 
and  from  the  light,  and  observe  the  image  of  the  flame  on  B. 
Account  for  the  inversion  of  the  image  and  its  varying  size, 
with  drawings  of  vertical  sections  to  illustrate  the  explanation. 

b.  Replace  C  with  a  sheet  of  paper  in  which  you  have  made 
a  small  hole  with  a  pin  or  the  point  of  your  pencil.     Hold  the 
paper  so  that  the  light  through  this  hole  will  form  an  image  on 
B,  and  note  the  effect  of  gradually  enlarging  the  hole  till  it  is 
2  cm.  or  more  in  diameter.     Describe  and  explain  the  effect  on 
the  image. 

c.  Make  a  few  small  holes  in  a  group  in  another  place  in  the 
paper,  and  hold  it  so  that  images  will  be  formed  by  them. 
Increase  the  number  of  holes  in  the  group  from  time  to  time, 
until  finally  there  are  many  of  them  very  near  together,  and 
observe  the  effect  on  the  images.     How  would  the  images  be 
affected  if  the  number  of  holes  were  indefinitely  increased  ? 

III.  To  find  the  relation  between  the  area  of  the  sur- 
face covered,  by  a  given  pencil  of  light  and  the  distance 
of  the  surface  from  the  source  of  the  light. 

a.  Place  A  30  cm.  and  B  60  cm.  from  the  flame,  turned  very 
low  and  placed  edgewise  at  the  end  of  the  optical  bench. 
(Measure  from  the  ./fame,  not  from  the  end  of  the  bench.) 
Measure  the  height  and  width  of  the  shadow. 

How  do  its  dimensions  compare  with  those  of  A  ? 

How  do  their  areas  compare  '/ 


PHOTOMETRY 


139 


If  A  were  removed,  the  light  which  now  falls  upon  it  would 
cover  how  many  times  as  great  an  area  at  B  ? 

How  would  the  intensity  of  illumination  upon  B  then  com- 
pare with  the  present  illumination  upon  A  ? 

b.  Repeat  the  preceding  with  A  30  cm.  and  B  90  cm.  from 
the  flame. 

c.  What  is  the  relation  between  the  distance  from  a  source 
of  light  and  the  area  covered  by  a  given  pencil  of  light,  from 
that  source  ?     Draw  a  figure  (in  perspective)  to  illustrate. 

d.  State  the  law  of  intensity  of  illumination,  and  show  how 
it  follows  from  your  answer  to  the  preceding  question. 


EXEECISE  48.     PHOTOMETKY 

References.  —  Hoadley,  447-449;  Carhart  and  Chute,  235- 
238  ;  Sanford,  pp.  333-335  ;  Jones,  Light,  6-10. 

DIRECTIONS  FOR  BUNSEN'S  PHOTOMETER 

Apparatus.  —  A  Bunsen's  photometer,  box  form  (Fig.  51) ; 
gas  jet ;  large  and  small  paraffine  candles  ;  blocks  for  supporting 
the  candles. 

I.  To  find  the  relation  between  the  intensity  of  illumi- 
nation and  the  distance. 

a.  Mount  a  small  candle  at  the  center  of  one  block  and  four 
of  the  same  size  on  another  block.  Place  the  single  candle 


o 


FIG. 


exactly  at  one  end  of  the  meter  rod  in  the  photometer  and  the 
center  of  the  group  of  four  candles  at  the  other  end.     With  the 


140  LIGHT 

lid  of  the  photometer  partly  closed  to  shut  out  external  light, 
move  the  sliding  piece  of  the  photometer  back  and  forth  be- 
tween the  lights  ;  and,  while  doing  so,  observe  in  the  mirrors 
the  changing  appearance  of  the  two  sides  of  the  oiled  spot  on 
the  screen. 

Is  the  side  of  the  spot  upon  which  the  stronger  light  falls 
the  brighter  or  the  darker  ?  Why  ? 

When  the  two  sides  of  the  spot  look  alike,  the  two  sides  of 
the  screen  are  equally  illuminated.  (The  candles  should  burn 
as  nearly  equally  as  possible.  To  make  them  do  so  it  may  be 
necessary  to  trim  the  wicks  occasionally.)  Find  the  position 
of  equal  illumination,  and  measure  the  distances  from  the  lights 
to  the  screen.  Let  D  denote  the  distance  of  the  four  candles 
and  d  the  distance  of  the  single  candle. 

b.  Without   moving   the   candles,  displace   the    screen  and 
make  a  new  adjustment.     If  the  new  position  does  not  differ 
by  more  than  1  cm.  from  the  first,  average  these  distances  with 
the  first  for  the  true  values  of  D  and  d.     The  results  of  differ- 
ent trials  may  differ  considerably.     This  is  principally  due  to 
the  fact  that  the  candles  do  not  burn  steadily.     If  the  differ- 
ences-are large,  take  the  average  of  several  trials. 

c.  Assuming  that  all  the  candles  give  equally  strong  light, 
the   illumination   on   the  side  of  the  screen  toward  the  four 
candles  is  four  times  as  great  as  it  would  be  if  illuminated  by 
a  single  candle  at  the  same  distance.     That  is,  the  illumination 
of  the  screen  by  a  single  candle  D  cm.  from  it  is  \  as  great  as 
it  would  be  at  a  distance  of  d  cm.     Or  thus :  for  distances  in 
the  ratio  D :  d,  the  intensities    of   illumination  due  to  equal 
sources  (or  the  same  source)  are  in  the  ratio  1 : 4.     Compute 
D :  d  and  (D  :  d)2,  and  compare  the  latter  with  the  ratio  (or  the 
reciprocal  of  the  ratio)  of  the  illuminating  power  of  a  light  at 
these  distances. 

How  should  these  ratios  compare?  Compute  the  per  cent 
of  error. 

Mention  probable  sources  of  error  in  the  experiment. 


PHOTOMETRY 


141 


II.    To  measure  the  candle  power  of  a  small  candle 
and  a  gas  jet. 

a.  Compare  the  intensity  (illuminating  power)  of   a  small 
candle  and  a  large  one.     Let  D  and  d  denote  respectively  the 
distance  of  the  large  and  the  small  candle  from  the  screen  for 
equal  illumination,  and  I  and  i  their  respective  illuminating 
powers.     Taking  the  larger  candle  as  the  standard,  the  candle 
power  of  the  small  one  is  i  -~-  I,  which  is  measured  by  (d  -f-  Z>)2. 
Make  two  or  more  trials,  and  record  as  indicated. 

b.  Find  the  candle  power  of  the  gas  jet  when  turned  to  a 
moderate  height,  by  comparing  it  with  the  large  candle.     Stand 
the  burner  on  a  block  so  that  the  flame  is  exactly  at  the  end 
of  the  rod  and  turned  flatwise  toward  the  screen. 

c.  If  there  is  time,  turn  the  flame  edgewise  toward  the  screen 
and  measure  its  candle  power  in  this  position. 

FORM  OF  RECORD  FOR  PART  II 


SOURCE  OF 
LIGHT 

D 

d 

d+D 

CANDLE  POWER 

(i  -r  /)=(<*  -5-  />)« 

small 

1 

CHI. 

cm. 

candle 

2 





Av. 









gas  jet 

1 





(flatwise) 

2 





etc. 

Av. 



^^^~" 

DIRECTIONS  FOR  THE  DIFFUSION  PHOTOMETER 

Apparatus.  —  A  diffusion  photometer;  optical  bench;  flat  gas 
jet;  large  and  small  paraffine  candles;  blocks  for  supporting 
candles. 

[The  diffusion  photometer  consists  of  two  cakes  of  paraffine  about  5  cm. 
square  and  1  cm.  thick,  separated  by  tin  foil  and  mounted  on  a  block 
(Fig.  52).  To  insure  equal  optical  properties,  they  must  be  cut  from  the 


142 


LIGHT 


same  piece  of  paraffine.  The  cakes  are  attached  to  the  foil  by  warming 
a  surface  of  each  till  it  begins  to  melt,  then  pressing  the  warmed  surfaces 
quickly  together  with  the  foil  between.  Attach  to  the  block  with  a  few 
drops  of  melted  paraffine.  The  photometer  in  this  form  gives  good  results 
if  the  room  is  well  darkened  and  there  are  no  lights  from  other  experi- 
ments to  interfere  ;  but  it  is  generally  better  to  inclose  the  paraffine  blocks 
in  a  cylinder  of  black  cardboard  with  a  hole  just  above  them  through 
which  they  can  be  observed,  as  shown  at  the  right  in  the  figure.] 

a.  Take  the  photometer  (the  mounted  paraffine  blocks)  in 
the  hand,  and  turn  it  about  at  different  angles  to  any  source  of 
light.  Observe  that  the  less  strongly  illuminated  side  always 
appears  the  darker.  Since  the  tin  foil  separating  the  blocks  is 


FIG.  52. 

opaque,  each  block  is   illuminated  only  from   its  own   side. 
They  have  the  same  tint  when  they  are  equally  illuminated. 

Darken  the  room  as  much  as  possible,  place  the  photometer 
on  the  optical  bench,  and  turn  the  latter  so  that  the  two  sides 
of  the  photometer  are  equally  illuminated  by  the  diffused  light 
of  the  room.  Mount  a  small  candle  at  the  center  of  one  block 
and  four  of  the  same  size  on  another.  Light  them  and  place 
them  on  the  optical  bench  near  each  end.  Move  the  photome- 
ter back  and  forth  between  the  lights  till  the  position  of  equal 
illumination  is  found.  If  necessary,  trim  the  candle  wicks  to 
make  them  burn  equally  before  making  this  adjustment.  Meas- 
ure the  distance  from  the  lights  to  the  photometer.  If  the 
supporting  blocks  are  of  uniform  length,  the  distance  from 
center  to  center  of  two  blocks  is  the  same  as  the  distance  from 
either  end  of  one  to  the  corresponding  end  of  the  other.  The 
latter  distance  is  the  more  convenient  one  to  measure. 


PHOTOMETRY  143 

b.  Without  moving  the  candles,  displace  the  photometer 
and  make  a  new  adjustment.  For  the  remainder  of  the  exer- 
cise, follow  the  directions  for  the  Bunsen  photometer,  begin- 
ning with  I  b. 

DIRECTIONS  FOB  B-UMFORD'S  PHOTOMETER 

Apparatus.  —  A  Bumford's  photometer,  consisting  of  a  screen 
of  white  cardboard  and  a  rod,  each  supported  vertically ;  large 
and  small  candles ;  blocks  for  supporting  the  candles ;  flat  gas 
jet  or  lamp ;  meter  rod. 

Figure  53  is  a  diagram  of  a  horizontal  section  showing  the 
arrangement  of  the  apparatus.  AB  represents  the  screen,  C 
the  vertical  rod,  i  and  /  the  two  lights  to  be  compared,  and 
s  and  S  the  shadows  cast  by  i  and  /  respectively.  The  rod 


FIG.  53. 

should  be  within  a  few  centimeters  of  the  screen,  and  the 
lights  placed  so  that  the  shadows  are  close  together,  but  not 
touching.  The  screen  should  be  turned  so  as  to  make  ap- 
proximately equal  angles  with  the  lines  si  and  SI. 

It  is  evident  that  each  light  illuminates  the  shadow  cast  by 
the  other ;  hence,  when  the  shadows  are  equally  dark  (that  is, 
equally  illuminated),  the  two  lights  give  equal  illumination  at 
their  respective  distances  from  the  screen.  The  room  must  be 
rather  dark,  or  other  sources  of  light  will  make  the  shadows 
too  faint  for  satisfactory  comparison. 


144  LIGHT 

Follow  the  directions  for  the  Bunsen  photometer,  with  the 
necessary  modifications  of  I.  a.  The  weaker  light  should  be 
placed  at  a  distance  of  30  to  40  cm.,  and  the  other  moved  back 
and  forth  till  the  position  is  found  for  which  the  shadows 
appear  equally  dark.  Measure  the  distances  SI  and  si  (denoted 
by  D  and  d  respectively).  Displace  /  and  make  a  new  trial. 

40.  Parallax.  —  Hold  a  pencil  out  at  arm's  length  toward 
some  object  at  a  distance  of  several  feet,  and  study  carefully 
the  apparent  change  in  the  position  of  the  pencil  with  respect 
to  the  object  beyond,  as  you  look  at  it,  first  with  one  eye,  then 
with  the  other  (keeping  one  eye  closed).  Discover  the  cause 
of  this  apparent  change  of  position. 

While  looking  steadily  with  one  eye  at  the  pencil,  move  the 
head  from  side  to  side,  and  observe  that  the  pencil  seems  to 
move  to  and  fro  in  front  of  the  object  beyond,  although  it  is 
really  held  at  rest.  Does  the  pencil  seem  to  move  in  the  same 
direction  as  your  eye,  or  the  opposite  ? 

Hold  the  pencil  nearer  the  object  (or  observe  its  apparent 
motion  with  respect  to  a  nearer  object),  and  continue  to  shorten 
the  distance  between  them  till  they  are  close  together.  How 
does  the  decrease  of  distance  affect  the  apparent  motion  of  the 
pencil  ? 

Study  in  the  same  way  the  apparent  motion  of  the  pencil 
with  respect  to  an  object  (a  finger)  held  between  it  and  the  eye. 
Observe  that  the  apparent  motion  of  the  finger  with  respect  to 
the  pencil  is  the  opposite  of  the  apparent  motion  of  the  pencil 
with  respect  to  the  finger. 

Continue  experimenting  till  yon  have  established  for  your- 
self the  following :  (1)  With  respect  to  the  more  distant  object, 
the  apparent  motion  of  the  nearer  one  is  opposite  to  the  motion  of 
the  eye;  (2)  as  the  distance  between  the  objects  diminishes,  the 
apparent  motion  of  either  with  respect  to  the  other  also  dimin- 
ishes; and  (3)  these  apparent  motions  disappear  when  the 
objects  are  brought  together. 


PLANE    MIRRORS  145 

Kecall  the  apparent  motions  of  near  and  distant  objects  when 
viewed  from  a  window  of  a  rapidly  moving  car;  and  find 
whether  they  agree  with  these  conclusions. 

The  apparent  displacement  of  an  object  caused  by  a  change 
in  the  position  of  the  observer  is  called  parallax.  The  fact 
that  parallax  between  two  objects  disappears  when  they  are 
brought  together  is  a  very  useful  one  in  locating  images ;  and 
for  this  purpose  it  will  be  necessary  to  remember  the  relation 
stated  just  above  in  italics. 


EXERCISE  49P     PLANE   MIRRORS 

References.  —  Hoadley,  450-455 ;  Carhart  and  Chute,  239- 
245 ;  Slate,  196-197. 

Apparatus.  —  A  rectangular  piece  of  plane  mirror  attached  to 
a  block  at  the  back  to  support  it  vertically ;  rule ;  protractor ; 
pins. 

[In  using  common  mirrors  for  the  study  of  images  an  error  is  involved, 
due  to  two  refractions  of  the  light  at  the  front  surface.  On  account  of 
these  refractions  the  distance  of  the  image  is  diminished  by  about  two 
thirds  the  thickness  of  the  mirror.  Hence  thin  mirrors  are  to  be  pre- 
ferred. The  error  is  reduced  one  half  if,  in  locating  the  image  by  parallax, 
the  object  that  is  made  to  coincide  with  the  image  is  viewed  through  an 
unsilvered  portion  of  the  glass.  The  error  will  be  entirely  avoided  if  the 
front  surface  of  a  piece  of  plate  or  w'indow  glass  is  used  as  the  reflecting 
surface.  The  image  will  be  quite  distinct  if  the  glass  is  backed  with  black 
paper  or  cloth.  Since  both  surfaces  reflect,  two  images  will  be  seen. 
The  rear  image  will  disappear  and  the  other  will  be  more  distinct  if  the 
back  of  the  glass  is  painted  black.  ] 

I.  To  find  the  position  of  a  point  image  in  a  plane 
mirror  by  sight  lines ;  and  to  find  the  relation  between 
the  position  of  the  point  and  its  image. 

a.  Draw  a  line  AB  (Fig.  54)  about  10  cm.  long  on  your 
record  sheet,  and  stand  the  mirror  with  the  edge  of  its  reflect- 
ing surface  on  this  line.  (The  reflecting  surface  of  a  common 
COLEMAN'S  PHY.  LAB.  MAN.  — 10 


14ti  LKJ1IT 

mirror  is,  of  course,  the  rear  surface.  If  unsilvered  glass  is 
used,  either  with  or  without  the  back  painted  black,  its  front 
surface  is  the  reflecting  surface.  If  the  back  surface  is  not 
painted  or  ground,  it,  too,  will  cause  an  image,  more  distant 
than  the  first,  but  it  is  to  be  disregarded.)  Stick  a  pin  verti- 
cally about  6  or  8  cm.  in  front  of 

B    the   mirror    (at   0).      You   are   to 

determine  accurately  the  position 
of  the  image  of  the  point  of  this 
pin,  seen  in  the  mirror.  Stick  a 
second  pin  near  the  mirror  and  a 
few  centimeters  to  one  side  of  0 
(at  (7),  and  another  pin  (at  D)  6  or 
8  cm.  from  C  and  exactly  in  line 

with  the  pin  at  C  and  the  image  of  the  pin  at  0.  The  pins 
should  be  as  nearly  vertical  as  possible,  and  the  eye,  in  sight- 
ing, placed  on  a  level  with  the  paper.  Draw  the  line  CD. 
The  image  of  the  pin  at  0  lies  somewhere  on  this  line  pro- 
duced. 

b.  Without  disturbing  the  position  of  the  mirror,  determine 
in  the  same  way  two  other  lines  directed  toward  the  image,  at 
different  angles  with  AB.  one  on  each  side  of  0.     If  the  image 
has  the  same  position  when  viewed  from  different  directions, 
these  lines  (and  all  others  similarly  drawn)  should  intersect  in 
the  point  which  is  the  position  of  the  image. 

Has  the  image  a  fixed  position  ? 

c.  Let  I  denote  the  position  of  the  image.     Draw  a  line  con- 
necting /  and  0. 

What  angle  does  this  line  make  with  AB  ?  (Measure  with 
the  protractor.) 

How  is  it  divided  by  AB  ? 

Describe  definitely  the  position  of  the  image  with  reference 
to  the  mirror  and  the  position  of  the  point  object. 

d.  Eepeat  the  experiment  with  the  point  0  taken  either  a 
greater  or  less  distance  from  the  mirror. 


PLANE    MIRRORS  147 

II.  To  find  the  relation  between  the  angle  of  incidence 
and  the  angle  of  reflection. 

a.  It  is  evident  that,  when  you  were  sighting  along  the  line 
DC  at  the  image,  the  light  by  which  you  saw  it  came  to  the 
eye  along  that  line,1  having  been  reflected  by  the  mirror  at  the 
point  where  DC  meets  it.  Call  this  point  N.  Draw  the  incident 
ray  ON  and  the  perpendicular  at  N.  Measure  with  the  protractor 
and  record  in  your  figure  the  angles  of  incidence  and  reflection. 
Repeat  this  construction  and  measurement  for  the  other  two 
reflected  rays. 

6.  Within  what  limits  (expressed  as  a  per  cent)  do  your 
results  agree  with  the  law  of  reflection  ? 

c.  What  is  meant  by  the  plane  of  the  angle  of  incidence  and 
of  the  angle  of  reflection  ?  What  is  the  plane  of  these  angles 
in  this  experiment  ? 

III.  To  find  the  relation  between  the   position   and 
appearance  of  an  object  and  its  image. 

a.  Write  your  name  on  a  piece  of  paper  and  look  at  its 
image  in  the  mirror.  Describe  and  account  for  its  appearance. 

6.  Draw  a  line  (AB)  on  your  record  sheet,  and  a  short  dis- 
tance below  it  draw  a  triangle  and  letter  the  vertices  (7,  Z>,  and 
E.  Leave  sufficient  space  above  AB  for  the  proper  location 
of  the  image  of  the  triangle.  Stand  the  mirror  on  AB,  and 
study  the  image  of  the  triangle,  noting  the  order  and  position 
of  its  sides  and  the  appearance  of  the  letters  at  the  vertices. 
The  vertices  of  the  triangle  may  be  located  experimentally  as 
in  I  a,  two  lines  for  each  vertex  being  sufficient,  or  by  the 
geometrical  construction  indicated  by  your  answers  to  I  c. 
Place  the  letters  C,  D,  and  E  in  the  drawing  of  the  image  just 
as  they  appear  in  the  mirror. 

1  CD  is,  strictly  speaking,  the  axis  of  the  pencil  or  cone  of  light  that 
enters  the  eye.  It  must  be  remembered  that  the  eye  locates  the  image  at 
the  point  from  which  this  cone  of  light  seems  to  come. 


148 


LIGHT 


EXERCISE   492.     PLANE   MIRRORS 

References.  —  The  same  as  for  Exercise  49]. 

I.  To  find  the  position  of  a  point  iunt^c  in  a  plane 
mirror  by  parallax;  and  to  find  the  relation  between  fhr 
position  of  the  point  and  its  image. 

Apparatus  (for  Parts  I  and  II).  — The  same  as  for  Exercise  ±9} 
with  the  exception  that  the  mirror  is  supported  with  a  free 
space  about  half  an  inch  high  under  it  (Fig.  55). 

a.  Draw  a  line  AB  about  10  cm.  long  on  your  record  sheet, 
leaving  a  blank  space  of  equal  width  above  it,  and  stand  the 
mirror  so  that  its  reflecting  surface  is  in  the  vertical  plane 
through  AB.     Adjust  it  accurately.     (The  reflecting  surface  of 

a  common  mirror  is,  of  course, 
the  rear  surface.  If  un  silvered 
glass  is  used,  either  with  or 
without  the  back  painted  black, 
its  front  surface  is  the  reflecting 
surface.  If  the  back  surface  is 
not  painted  or  ground,  it  too  will 
cause  an  image,  more  distant 
than  the  first,  but  it  is  to  be  disregarded.)  Stick  a  pin  verti- 
cally 6  or  8  cm.  in  front  of  the  mirror.  Place  a  second  pin 
behind  the  mirror,  and  find  by  parallax  (Art.  40)  the  position 
in  which  the  portion  of  it  that  is  seen  under  the  mirror  (the 
eyes  being  nearly  on  a  level  with  the  table)  fits  accurately  to 
the  portion  of  the  image  of  the  pin  in  front  that  is  seen  at  the 
same  time  in  the  mirror.  Use  both  eyes,  and  move  the  head 
from  side  to  side.  When  the  correct  position  is  found,  the  pin 
and  the  image  fit  from  all  points  of  view.  Let  Oj  denote  the 
position  of  the  pin  in  front  and  7X  the  position  of  its  image. 

b.  Place  the  pin  at  a  different  distance  from  the  mirror,  and 
again  locate  the  image.     Let  02  and/2  denote  the  position  of 
object  and  image  respectively. 

c.  Follow  the  directions  of  I  c  of  Exercise  49P 


FIG.  55. 


PLANE    MIRRORS  149 

II.  To  find  the  relation  between  the  position  and  ap- 
pearance of  an  object  and  its  image. 

Follow  the  directions  of  Part  III  of  Exercise  49i  locating 
the  vertices  of  the  triangle  by  parallax  or  by  geometrical  con- 
struction. 

III.  To   find  the  relation    between  the  angle  of  in- 
cidence and  the  angle  of  reflection.1 

Apparatus.  —  Reflection  apparatus  (Fig.  56). 

a.  Hold  a  finger  over  the  first  hole  on  either  side  of  the 
middle  hole  of  the  reflection  apparatus  (Fig.  56).     (If  the  room 
is  darkened  for  other  experiments,  a  lighted  candle  may  be 
used  instead  of  the  finger.) 

By  trial  find  the  hole 
through  which  the  image 
of  the  covered  (or  illumi- 
nated) hole  can  be  seen 
in  the  mirror.  Place  the  FlG-  56< 

finger  over  the  next  hole,  and  repeat  the  observation.  Try  all 
the  holes  on  one  side  of  the  center  in  the  same  way ;  and  in 
each  case  observe  how  the  arc  of  the  rim  between  the  middle 
hole  and  the  covered  hole  compares  in  length  with  that  between 
the  middle  hole  and  the  one  through  which  the  image  is  seen. 

How  do  the  angles  subtended  at  the  center  of  the  mirror  by 
these  arcs  compare  ?  These  angles  at  the  center  are  the  angles 
of  incidence  and  reflection  respectively,  for  the  line  from  the 
middle  hole  to  the  center  of  the  mirror  is  perpendicular  to  it. 
State  the  relation  that  holds  for  these  angles. 

b.  Draw  a  diagram  showing  an  incident  and  the  reflected  r;i y 
and  the  perpendicular  to  the  mirror  at  the  point  of  incidence;  and 
indicate  the  angles  of  incidence  and  reflection  by  i  and  r. 

What  is  meant  by  the  plane  of  these  angles  ?  What  does  the 
experiment  show  concerning  these  planes  ? 

1  This  part  may  be  omitted  and  the  law  of  reflection  deduced  geometri- 
cally from  the  results  of  Part  I. 


150  LIGHT 

EXERCISE   50.     MULTIPLE   IMAGES 

References. — Hoadley,  456-458;  Carhart  and  Chute,  246-247; 
Slate,  200 ;  Jones,  Light,  22-24. 

Apparatus.  —  Two  plane  mirrors  with  support  at  the  back  to 
hold  them  in  a  vertical  position ;  bits  of  colored  glass  or  paper  ; 
rule ;  candle  ;  an  object  with  distinguishable  right  and  left  sides 
and  front  and  back.  A  small  block  with  some  printed  matter 
pasted  on  one  side  (called  the  front)  will  serve. 

I.  To  study  tlie  formation  of  images  by  two  mirrors 
at  right  angles. 

a.  Stand  the  mirrors  at  right  angles  to  each  other  with  a 
vertical  edge  of  each  together.     To  do  this,  adjust  the  mirrors 
so  that  the  image  of  each  is  in  the  same  plane  as  the  mirror 
itself.     Place  the  block  between  the  mirrors  and  turn  it  so 
that  the  printing  is  visible  in  both  of  them.     Observe  the  ap- 
pearance of  the  printing  in  each  of  the  images.     Note  also  the 
location  of  the  images  with  respect  to  the  mirrors,  the  images 
of  the  mirrors,  and  the  object.     Study  these  matters  further  as 
you  move  the  block  about  between  the  mirrors. 

Draw  a  diagram  of  what  is  seen  with  the  block  in  one  posi- 
tion, representing  the  block  by  a  rectangle  with  the  vertices 
lettered  A,  B,  C,  and  D.  (To  locate  the  image  of  a  point,  draw 
a  perpendicular  from  the  point  to  the  line  representing  the 
mirror,  and  extend  it  an  equal  distance  beyond.)  Letter  the  ver- 
tices of  the  images  to  correspond.  The  correct  lettering  of  the 
vertices  will  show  which  images  are  reversed  and  which  are  not. 

Any  corner  of  the  object  and  its  three  images  form  the  ver- 
tices of  what  geometrical  figure  ? 

b.  Cover  first  one  of  the  mirrors  then  the  other  with  a  sheet 
of  paper,  and  observe  which  of  the  images  remain.     Which 
images  are  formed  by  one  mirror  only  ?    Which  by  both?    Num- 
ber the  images  in  your  diagram,  and  answer  by  reference  to 
these  numbers. 


MULTIPLE    IMAGES  151 

c.  The  light  by  which  you  see  the  image  that  is  formed  by 
both  mirrors  is  reflected  from  the  mirror  in  which  the  image  is 
seen,  after  a  previous  reflection  from  the  other  mirror.  After 
the  first  reflection,  the  light  falls  upon  the  second  mirror  at  pre- 
cisely the  same  angle  that  it  would  if  it  came  from  the  image 
in  the  first  mirror.  This  will  be  understood  from  a  study  of 
Fig.  57;  in  which  O  is  a  point  object  and  II  its  image  in  the 


mirror  AB.  A  pencil  of  light  from  0,  after  reflection  from  AB, 
will  seem  to  come  from  the  image  Il9  and  will  fall  upon  the 
mirror  CD  and  be  reflected  by  it  just  as  if  it  did  come  from  /,. 
Hence  J2,  the  image  in  CD  that  is  formed  by  reflection  from 
CD  after  reflection  from  AB,  is  most  simply  located  by  first 
locating  I19  then  treating  it  as  the  object  whose  image  is  7,,. 
In  this  sense  J2  is  an  image  of  the  first  image  7j. 

Use  this  method  in  locating  images  in  parallel  mirrors  or  in 
mirrors  at  any  angle. 

II.  To  study  the  formation  of  images  by  two  mirrors 
at  an  angle  of  60*. 

a.  Stand  the  mirrors  at  an  angle  of  60°.  To  do  this,  turn 
the  mirrors  gradually  together  till  each  is  in  line  with  tin- 
second  mirror  image  in  the  other  mirror.  Plar<-  tin-  M"<-k 


152 


LIGHT 


between  the  mirrors  and  study  its  images  as  before.  Draw 
a  diagram  of  the  observed  arrangement  of  object  and  images, 
and  indicate  corresponding  vertices  by  lettering. 

b.  Place  a  few  small  objects  (such  as  bits  of  colored  glass 
or  paper)  between  the  mirrors  and  near  the  vertex  of  the  angle 
between  them.  Move  the  objects  into  different  positions,  and 
observe  the  symmetry  of  the  different  patterns  formed  by  the 
objects  and  their  images.  This  illustrates  the  principle  of  the 
kaleidoscope.  Draw  one  or  two  of  the  patterns. 

If  a  kaleidoscope  is  provided,  look  through  it. 

III.  To  study  the  formation  of  images  by  two  parallel 
mirrors. 

a.  Stand  the  mirrors  parallel  to  each  other  and  a  few  centi- 
meters apart.     Place  the  block  between  them,  and  observe  its 
images  in  each  mirror.     Draw  a  diagram  of  the  block  and  mir- 
rors and  their  images. 

b.  Light  the  candle  and  place  it  between  the  parallel  mir- 
rors.     Why  are  there  more  images  of  the  candle  than  there 
were  of  the  block  ? 

c.  Hold  the  candle  near  one  of  the  mirrors,  and  look  at  its 
image  obliquely  to  the  mirror.     What  fainter  images  are  seen 
near  the  principal  one  ?     Account  for  them. 

Discussion.  —  a.    Draw  a  figure  of  two  mirrors  at  right  angles, 

showing  the  images  of  a 
point  object  and  a  ray 
from  the  object  to  the  eye 
for  each  image.  Place 
both  the  object  and  the 
eye  at  unequal  distances 
from  the  mirrors.  Locate 
the  images  by  the  method 

described  in   I  c.      Draw 
t 

the  last  part  of   the  ray 
FIG.  58.  first,  from  the  eye  to  the 


THE    CONCAVE    MIRROR  153 

mirror,  in  the  direction  of  the  image  from  which  it  seems  to 

come;  then,  if  this  image  is  a  second  image,  draw  the  next 

part    of    the    ray   to    the        rr 

other  mirror,  in  the  direc-      *    ®^ 

tion  of  the  corresponding     Tr         -vr--— 

first  image.  Vx-^      xvx 

b.  Draw  a  similar  figure     - 
for  two  mirrors  at  an  angle 

of  60°,  nsing  Fig.  58  as  a     . 

model.      In    constructing 

the  figure,  locate  first  the  .»"'"'.'' 

point     object    (unequally  ^"'    ^' 

distant  from  the  mirrors),       *  &'      ^-' 

then  the  two  first  images,     ~J"~~' 

,_,  I    9  FIG.  59. 

ii   and   /!,   then   the    two 

second  images,  iz  and  I2,  and  lastly  the  third  image,  is  or  J3, 
according  to  the  position  of  the  eye.  It  will  be  a  help  in  the 
construction  to  make  use  of  the  fact  that  the  object  and  its 
images  lie  on  the  circumference  of  a  circle. 

c.  Draw  a  figure  for  parallel  mirrors  and  a  point  object. 
Draw  rays  from  the  object  to  the  eye  for  a  few  of  the  images. 

EXERCISE  51.     THE   CONCAVE   MIRROR 

References.— Hoadley,  460-466;  Carhart  and  Chute,  249- 
255 ;  Slate,  201  and  203 ;  Jones,  Light,  32-35. 

Apparatus.  —  Optical  bench  or  meter  rod  ;  mounted  candle ; 
concave  mirror ;  cardboard  screen  with  hole  5  or  6  cm.  in 
diameter. 

[It  will  be  convenient  to  have  the  mirror  mounted  uniformly  with 
the  screens,  lenses,  etc.'  (See  suggestions  under  Exercise  47.)  The 
hole  in  the  screen  should  be  at  the  same  height  as  the  central  portion 
of  the  mirror;  so  that  light  from  more  distant  objects,  after  passing 
through  the  hole,  will  fall  upon  the  mirror  and  be  reflected  by  it  to 
the  rear  surface  of  the  screen.] 


154  LIGHT 

I.    To  find  the  focal  length  of  a  concave  mirror;  and  to 
study  the  real  images  formed  by  it. 

a.  Hold  the  mirror  in  the  sunlight  and  facing  the  sun,  and 
focus  the  light  on  a  piece  of  paper.      To  do  this,  move  the 
paper  to  and  fro  till  the  place  is  found  where  the  spot  of  light 
is  the  smallest.     This  small,  round  spot  of  light  is  a  real 
image  of  the  sun.     Its  position  is  called  the  principal  focus  of 
the  mirror.     (See  definition  in  text.) 

b.  Kaise  a  window  and  turn  the  optical  bench  so  that  the 
mirror,  when  placed  upon  it,  will  face  some  distant  object. 
Focus  the  image  of  this  object  upon  the  screen  by  placing  the 
screen  upon  the  bench  and  adjusting  its  distance  from  the  mir- 
ror till  the  image  is  sharply  defined  upon  it.     A  screen  with- 
out a  hole  may  be  used  by  placing  it  a  little  to  one  side  of  a 
direct  line  between  the  object  and  the  mirror.     If  the  object 
is  at  a  distance  of  200  ft.  or  more,  the  image  is  very  nearly  at 
the  principal  focus.     Try  to  discover  a  reason  for  this  while 
studying  conjugate  foci  in  Part  III.     Measure  the  distance  of 
the  image  from  the  mirror.     Assuming  the  image  to  be  at  the 
principal  focus,  this  distance  is  the  focal  length  (/)  of  the 
mirror. 

c.  Move  the  screen  a  little  to  one  side,  leaving  it  in  the 
plane  of  the  image  and  near  it  so  as  to  mark  its  position. 
Now  stand  at  a  distance  of  about  2  m.  from  the  mirror  and 
very  nearly  in  line  between  it  and  the  object,  and  look  at  the 
screen.     You  should  see  the  image  of  the  distant  object  beside 
the  screen  and  in  the  air  where  it  really  is ;  and,  when  seen 
thus,  it  will  be  very  distinct.      The  natural  tendency  is  to 
look  in  the  mirror  for  the  image ;    and  it  will  seem  to  be 
in   the   mirror   and  will   appear   blurred   if   the  eyes  are   so 
directed. 

d.  Try  viewing  the  image  directly  from  different  positions. 
Can  you  see  it  from  as  many  different  directions  as  you  can 
when  it  is  caught  upon  a  screen  ?     Explain. 


THE   CONCAVE    MIRROR  155 

II.  To  find  the  center  of  curvature  of  the  mirror. 

a.  Close  the  window  and  blinds,  or  carry  the  apparatus  to  a 
darker  room.     Light  the  candle  and  place  it  upon  the  bench  at 
one  end  and  the  mirror  at  the  other.      Stand  about  2  m.  in 
front  of  the  mirror,  and  with  the  eye  on  a  level  with  the  can- 
dle, look  for  real  images  of  it  in  front  of  the  mirror.     The  one 
that  is  the  largest  and  the  most  distant  from  the  mirror  is 
much  the  brightest,  and  is  the  one  to  be  studied.     The  others 
are  due  to  multiple  reflections  within  the  mirror.      A  faint, 
unmagnified,  virtual  image  will  also  be  visible.     It  is  due  to 
reflection  from  the  front  surface  of  the  mirror,  which  is  a  plane 
surface. 

b.  Move  the  candle  on  the  bench  till  the  image  is  directly 
above  it.     If  you  have  difficulty  in  seeing  the  image  where  it 
actually  is,  locate  it  by  catching  it  upon  the  screen.     If  the 
candle  is  adjusted  to  the  proper  height,  the  tip  of  the  flame 
and  its  image  can  be  made  to  coincide.     They  are  then  at  the 
center  of  curvature  of  the  mirror.     Why?     Make  this  adjust- 
ment and  measure  the  distance  of  the  candle  from  the  mirror. 
This  is  the  radius  of  curvature  (r)  of  the  mirror. 

c.  What  simple  relation  (within  2%  or  3%)  do  you  discover 
between  r  and  /  ? 

III.  To  study  the  relation  between  the  position  of  the 
object  and  the  size  and  position  of  its  real  or  virtual, 
image. 

a.  Starting  with  the  candle  at  the  center  of  curvature,  move 
it  slowly  across  the  room  as  far  as  you  can  from  the  mirror, 
and  observe  the  simultaneous  movement  of  the  image  and 
change  in  its  size. 

Describe  this  motion  of  the  image  with  reference  to  the 
center  of  curvature  and  the  principal  focus. 

Where  is  the  image  with  reference  to  the  principal  focus 
when  the  object  is  most  distant? 


156  LIGHT 

What  would  be  the  final*  position  of  the  image  if  the  object 
were  carried  farther  away  indefinitely  ? 

6.  Again  starting  with  the  candle  at  the  center  of  curvature, 
move  it  slowly  toward  the  principal  focus  ;  and  study  the  image 
as  before,  except  that  it  is  to  be  caught  upon  the  screen.  Con- 
tinue till  the  image  is  focused  upon  the  wall  or  as  far  away  as 
it  can  be  seen. 

Where  is  the  candle  now  with  reference  to  the  principal  focus 
and  the  center  of  curvature? 

If  it  were  moved  up  to  the  principal  focus,  where  would  the 
image  be  ? 

Whenever  a  real  image  is  formed,  the  rays  from  any  point 
of  the  object  converge  to  the  corresponding  point  of  the  image 
after  reflection  from  the  mirror.  Do  the  rays  after  reflection 
become  more  or  less  convergent  as  the  candle  is  moved  from 
the  center  of  curvature  toward  the  principal  focus  ? 

Are  they  convergent,  divergent,  or  parallel  when  the  candle 
is  at  the  principal  focus  ? 

c.  Move  the  candle   from  the   principal  focus  toward  the 
mirror ;  and  at  the  same  time  study  the  image,  which  is  now 
virtual  and  must  be  looked  for  in  the  mirror.     Describe  its 
change  of  size  and  position  during  the  motion. 

Are  the  rays  convergent  or  divergent  after  reflection  ? 

d.  With  the  mirror  at  one  end  of  the  bench  and  the  screen 
at  the  other,  place  the  candle  so  that  its  image  is  distinct  upon 
the  screen.     Interchange  the  positions  of  the  candle  and  the 
screen,  placing  each  exactly  where  the  other  was  before.     Is 
the  image  now  distinct  upon  the  screen  ? 

What  property  of  conjugate  foci  does  this  illustrate? 

Discussion. — The  point  of  convergence  of  a  pencil  of  rays 
after  reflection  from  the  mirror  (or  the  point  from  which  the 
pencil  seems  to  come,  if  it  is  diverging  after  reflection)  is  de- 
termined by  the  intersection  of  any  two  rays  of  the  pencil.  Now 
there  are  certain  rays  whose  directions  after  reflection  are  very 


PHENOMENA   DUE    TO   REFRACTION  157 

simply  determined  without  the  construction  of  angles  of  inci- 
dence and  reflection ;  and,  on  the  ground  of  simplicity,  these 
rays  should  always  be  used  in  drawing  figures  showing  the 
formation  of  images  by  concave  (or  convex)  mirrors.  They  are 
the  following :  — 

(1)  The  ray  (from  the  chosen  point  of  the  object)  parallel  to 
the  principal  axis  ;  which  passes  through  the  principal  focus 
(F)  after  reflection. 

(2)  The  ray  through  F\  which  is  reflected  parallel  to  the 
principal  axis. 

(3)  The  ray  through  the  center  of  curvature  (0) ;  which  is 
reflected  back  along  the  same  path. 

With  an  arrow  as  object,  draw  figures  showing  the  formation 
of  the  image  by  a  concave  mirror,  —  (1)  when  the  object  is  be- 
yond C;  (2)  when  it  is  between  F  and  (7;  (3)  when  it  is  a 
little  nearer  the  mirror  than  F;  (4)  when  it  is  very  near  the 
mirror. 

EXERCISE  52.     PHENOMENA  DUE  TO  REFRACTION 

References.  —  Hoadley,  467-470  and  474-476;  Carhart  and 
Chute,  256-259  and  261-265;  Slate,  207-209  and  213-214; 
Jones,  Light,  42. 

I.  To  study  the  apparent  displacement  of  objects  under 
water,  due  to  refraction  at  the  surface. 

Apparatus.  —  Glass  jar  (preferably  rectangular)  ;  bit  of  tin  or 
other  small  object  that  sinks;  beaker;  jar  of  water;  mop 
cloth. 

a.  Place  the  bit  of  tin  (or  other  small  object)  in  the  empty 
jar  at  the  side  farthest  from  you,  and  hold  the  eye  a  very  little 
too  low  to  see  it  over  the  edge  of  the  jar.  Keep  the  eye 
steadily  in  this  position  while  your  companion  slowly  pours 
water  into  the  jar,  being  careful  not  to  displace  the  tin,  and 
observe  any  change  in  its  apparent  position. 


158 


LIGHT 


Is  the  direction  of  the  apparent  displacement  horizontal, 

vertical,  or  oblique  ? 

The  light  that  enters  the  eye  is  bent  (refracted)  on  passing 

from  the  water  into  the  air,  as  shown  in  Fig.  60.     The  object 

appears  to  be  on  the  line 
determined  by  the  direction 
of  the  ray  as  it  enters  the  eye. 
Its  position  on  that  line  is 
the  point  from  which  the 
diverging  cone  of  light  that 
enters  the  eye  seems  to  come. 
This  point  is  determined  from 
the  experiment  by  noting  the 

direction  of  the  apparent  displacement.     Draw  a  figure  similar 

to  Fig.  60,  and  complete  it,  showing  the  real  and  the  apparent 

position  of  the  tip. 

b.  Look  at  the  piece  of  tin  through  the  surface  of  the  water 
while  disturbing  the  water  with  your  finger.      Describe  and 
explain  the  effect  of  the  motion  of 

the  surface. 

c.  Put    your    pencil    into    the 
water   obliquely   half    its   length, 
and  note  its   appearance   as  seen 
through  the  surface  of  the  water. 
Draw  a  figure  similar  to  Fig.  61, 
and     complete    it,     showing    the 
apparent  position  of   the  portion 
of  the  pencil  under  water. 

Study  further  the  direction  of 
apparent  displacement  by  noting 
the  appearance  of  the  pencil  when 
held  vertical  and  partly  under 
water.  This  should  leave  no  possibility  of  doubt.  (The  illus- 
trations in  many  text-books  are  very  inaccurate  and  misleading 
on  this  point.) 


FIG.  61. 


PHENOMENA   DUE   TO   REFRACTION  159 

d.    Look  vertically  into  the  jar  of  water,  and  observe  the 
apparent   elevation   of    the    bottom   above  the   level   of    the 
table   top.      Figure    62   represents    a  pencil  of 
light  from  a  point  of  the  bottom.     Draw  and 
complete  the  figure   so  as   to   explain  the 'ap- 
parent elevation  of  the  bottom.     Represent  the 
apparent  path  of  light  by  a  dotted  line. 

II.  To  study  the  apparent  displacement 
of  objects  seen  through  plate  glass. 

Apparatus.  —  A  rectangular  piece  of  thick 
plate  glass,  preferably  ground  and  polished  on 
two  opposite  edges. 

a.  Hold  the  piece  of  plate  glass  up  before 

you,  and  hold  your  pencil  behind  it  so  as  to  see  part  of  the 
pencil  above  the  glass  and  part  through  it.     Turn  the  glass 

from    side   to    side   about  the 
pencil  as  an  axis,  and  note  the 

effect  on  the  apparent  position 

of  the  part  of  the  pencil  seen 
through  it. 

What  is  the  direction  of  the 
rays  through  the  glass   when 
FIG.  63.       \  there   is   no    apparent    lateral 

displacement  of  the  pencil  ? 

Figure  63  represents  a  section  taken  at  right  angles  to  the 
pencil.  Complete  it  so  as  to  explain  the  apparent  lateral  dis- 
placement of  the  pencil,  representing  its  apparent  position  by 
a  dotted  circle. 

b.  Compare  the  real  thickness  of  the  glass  with  the  apparent 
thickness  when  you  look  through  it. 

Compare  its  width  with  its  apparent  width  as  you  look 
through  it  from  edge  to  opposite  edge.  Draw  a  figure  explain- 
ing the  observed  effect. 


o 


160  LIGHT 

III.  To  find  the  path  of  a  ray  ^  of  light  passing  from 
air  through  a  triangular  prism  and  into  the  air  again. 

Apparatus. — A  triangular  glass  prism  with  flat  ends;  pins. 
Stick  two  pins  vertically  in  your  record  sheet  about  two- 
thirds  as  far  apart  as  the  width  of  a  side  of  the  prism,  and 
stand  the  prism  between  them,  as  shown  in  Fig.  64.  With 
the  eye  in  position  to  see  the  two  pins 
in  line  (the  farther  one  through  the 
faces  AB  and  AC),  stick  a  third  pin 
behind  the  prism  and  a  fourth  in  front 
of  it,  all  about  equally  spaced  and  all 
apparently  in  a  straight  line  when  the 
two  farther  ones  are  seen  through  the 
prism.  Draw  an  outline  of  the  base  of 
the  prism ;  remove  it  and  draw  straight 
lines  connecting  the  positions  of  the 
pins.  This  broken  line  is  the  path  of  a  ray  from  the  farthest- 
pin  to  the  eye. 

Draw  perpendiculars  to  the  refracting  surfaces  at  the  point 
of  entrance  and  emergence  of  the  ray,  and  indicate  the  angles 
of  incidence  and  refraction  of  the  rays  and  the  angle  of  devia- 
tion. 

State  the  direction  of  deviation  (toward  or  from  the  per- 
pendicular) in  passing  into  and  out  of  the  prism. 

41.  The  Sine  of  an  Angle.  —  In  a  right  triangle,  the  ratio  of  a 
leg  to  the  hypotenuse  is  called  the  sine  of  the  angle  opposite 
the  leg.  Thus,  in  the  right  triangle  .AB C,  BC:  BA  is  the  sine 
of  angle  A  and  CA:  BA  is  the  sine  of  angle  B.  The  usual 
form  of  expression  is 

sine  A  =  BC  +  BA,  sine  B  =  CA  -+-  BA. 

The  three  right  triangles  ABC,  AB'C',  and  AB"C'f  are  simi- 
lar; hence 

BC:BA  =  B'C' :  B'A  =  B"C"  :B"A  =  sine  A. 


INDEX   OF   REFRACTION 


161 


the   sine   of  an  angle   is 


From  which  it   will   be  seen  that 

independent  of  the  size  of  the 

triangle.      It    does,    however, 

depend  upon  the  size  of  the 

angle ;  increasing  as  the  angle 

increases   (up  to  90°),   but  not 

proportionally.      The     student 

should  assure  himself   of  the 

truth    of    this    statement    by 

drawing    angles    of    different 

sizes  and  roughly  comparing  the  angles  and  their  sines. 

42.  Index  of  Refraction.  —  The  index  of  refraction  of  a  sub- 
stance is  defined  as  the  ratio  of  the  sine  of  the  angle  of  inci- 
dence to  the  sine  of  the  angle  of  refraction  when  light  enters 
the  substance  from  a  vacuum.  That  is :  — 

Index  of  refraction  =  sine  i :  sine  r  = :  —  • 

AO   CO 

If  the  distances  AO  and  OC,  along  the  incident  and  refracted 
rays  respectively,  are  taken  equal,  this  expression  for  the  index 
of  refraction  reduces  to :  — 

Index  of  refraction  =  sine  i  :  sine  r  =  AB :  CD. 

The  methods  employed  in  elementary 
physics  are  not  accurate  enough  to  detect 
any  difference  between  the  index  of  refrac- 
tion from  a  vacuum  into  a  substance  and 
from  air  into  the  same  substance.  Hence 
such  differences  may  be  neglected. 

43.    Index    of    Refraction    by    Apparent 
Depth.  —  In  Fig.  67,  EOB  represents  a  sec- 
tion  of  a  narrow   cone   of   rays   from   a 
point  0  on  the  surface  of  some  substance.    OA,  the  axis  of  the 
cone,  is  perpendicular  to  the  opposite  surface,  at  which  the  light 
COLEMAN'S  PHY.  LAB.  MAN.  — 11 


FIG.  66. 


162 


LIGHT 


passes  into  the  air.  It  was  by  such  a  cone  of  rays  that  you 
observed  the  apparent  elevation  of  the  bot- 
tom of  the  jar  of  water  in  Part  I  c?,  of  the 
last  exercise.  The  figure  applies  also  to  Part 
II  b.  The  angles  of  incidence  and  refraction 
in  the  figure  are  taken  for  light  traveling  in 
the  opposite  direction. 

NM  is  the  perpendicular  to  the  surface  at 
BJ  hence  it  is  parallel  to  AO. 

Hence,  also,  angle  ACB  =  angle  NED  and 
angle  AOB  =  angle  OEM. 

Index  of  refraction  =  sine  i  :  sine  r  = : =— — 

CB   OB      CB 

For  a  narrow  pencil  of  light  (such  as  enters  the  eye)  the 
error  is  altogether  inappreciable  when  we  write  OB  =  OA  and 
CB=  CA-,  hence 

Index  of  refraction  =^|^^4  _ 

CB      CA     apparent  depth 


FIG.  67. 


EXEECISE   53.     INDEX  OF  KEFRACTION 

References.  —  Hoadley,  467-471 ;  Carhart  and  Chute,  256-260. 

I.  To  find  the  index  of  refraction  of  a  piece  of  plate 
glass. 

Apparatus.  —  Piece  of  thick  plate  glass  with  at  least  one 
straight,  polished  edge ;  pins  ;  metric  rule. 

First  Method.  —  a.  In  this  experiment  the  numerical  values 
are  obtained  from  the  geometrical  figures ;  hence  they  should 
be  constructed  as  accurately  as  possible  with  a  sharp  pencil. 

Draw  a  straight  line  AB  about  8  cm.  long  on  your  record 
sheet,  and  lay  the  piece  of  plate  glass  flat  on  the  sheet  with  a 
polished  edge  on  this  line  (Fig.  68).  Stick  two  pins  vertically 
against  the  edges  of  the  glass  at  C  and  D.  If  a  scratch  or 
other  mark  has  been  made  across  an  edge  of  the  glass,  use  it 


INDEX   OF   REFRACTION 


163 


instead  of  the  pin  at  C.     The  line  CD  must  be  quite  oblique 

to  AB.     With  the  eye  on  a 

level  with  the  paper,  move 

it  into  line  with  the  pin  at 

D  and  the  image  of  the  pin 

at  C,  seen  through  the  glass; 

and    place    a    third    pin    in 

this   line   at  E,  5  or  6   cm. 

from   D.      (The    three    pins  Fm' 68> 

must  seem  to  stand  exactly  in  a  straight  line.) 

b.  Eemove  the  glass  and  draw  straight  lines  connecting  (7, 
D,  and  E.     This  is  the  path  by  which  light  came  to  the  eye 

from  C.  Draw  a  line  (NM)  per- 
pendicular to  AB  at  D.  (For  a 
right  angle  to  measure  by,  fold 
a  piece  of  paper  so  that  the  two 
parts  of  a  straight  edge  of  it  fit 
B  accurately  together.)  Lay  off  on 
DC  and  DE  equal  distances  DF 
and  DG  (not  less  than  5  cm.) ; 
and  from  F  and  G  drop  perpen- 
diculars to  NM  (FH  and  GI). 
Measure  FH  and  GI  as  accurately 
as  possible,  and  compute  GI:  FH.  This  is  the  index  of  refrac- 
tion of  the  plate  glass  (Art.  42). 

c.  Repeat  the  experiment  with  the  point  D  taken  nearer  the 
corner  of  the  glass  (if  possible)  so  that  the  angles  of  incidence 
and  refraction  will  be  larger. 

d.  Compute  the  per   cent   of   difference   between  the  two 
values  of  the  index  of  refraction.     This  difference  is  due  to 
experimental  errors  or  to  errors  of  construction  and  measure- 
ment, and  should  not  exceed  3r%  or  4%. 

Second  Method.  —  The  piece  of  plate  glass  should  have  a 
scratch  or  other  mark  (A,  Fig.  70)  across  the  edge  opposite  to 
a  polished  edge.  Paste  a  strip  of  paper  (B)  on  a  side  of  tin- 


M 

FIG.  09. 


164 


LIGHT 


/N 
FIG.  70. 


glass  with  a  straight  edge  of  it  in  line  with  the  perpendicular 
(AN)  from  the  mark  to  the  polished  edge  CD.  Hold  the 
plate  on  a  level  with  the  eyes,  and  look  with  both  eyes  at 
the  mark  through  the  edge  CD,  along 
the  line  NA;  and  at  the  same  time 
through  the  air  at  the  sharp  point  of 
your  pencil,  held  against  the  glass  at 
the  edge  of  the  strip  of  paper.  Adjust 
the  position  of  the  pencil  by  parallax, 
shifting  the  eyes  a  little  to  right  and 
left,  till  the  pencil  point,  seen  through 
the  air,  coincides  with  the  image  of  the 
mark,  seen  through  the  glass.  When 
the  correct  adjustment  is  secured,  mark  on  the  paper  strip  the 
position  (E)  of  the  pencil.  Make  at  least  two  or  three  trials. 
Measure  accurately  the  width  of  the  glass  and  its  apparent 
width  when  viewed  through  it ;  and  compute  its  index  of 
refraction  (Art.  43). 

II.   To  find  the  index  of  refraction  of  water. 

Apparatus.  —  Battery  jar ;  board,  with  crosspiece  to  support 

it   on   the   jar;    pins;    metric   rule;  

mop  cloth. 

First  Method.  —  a.  Stick  a  pin 
perpendicularly  into  the  board  at  C 
(Fig.  71)  and  place  the  board  in  the 
jar.  Fill  the  jar  nearly  full  of  water, 
and  stick  a  pin  perpendicularly  into 
the  board  near  each  side  of  the  jar 
(at  A  and  B)  exactly  at  the  water 
level.  The  pins  must  mark  as  accu- 
rately as  possible  the  general  level 
of  the  water,  not  the  raised  edge  in 
contact  with  the  board.  Stick  a  pin  at  Z>,  toward  the  opposite 
side  of  the  jar  from  C  and  within  1  cm.  of  the  water,  but  not 


FFC.  71. 


INDEX  OF  REFRACTION  165 

touching  it.  Look  with  one  eye  along  the  surface  of  the  board, 
and  place  a  pin  at  E  in  line  with  D  and  the  apparent  position 
of  C,  so  that  the  three  pins  appear  to  be  exactly  in  a  straight 
line. 

Remove  the  board,  wipe  it  dry,  and  see  that  the  pins  are 
stuck  in  firmly.  Push  the  pins  through  a  sheet  of  paper,  and 
spread  the  paper  smoothly  over  the  board.  Being  careful  not 
to  let  the  paper  slip  about,  draw  with  a  sharp  pencil  a  straight 
line  between  the  pins  at  A  and  B  and  another  between  E  and 
D.  Continue  the  latter  till  it  meets  the  line  AB,  and  call  the 
point  of  intersection  0.  Draw  a  straight  line  connecting  0 
and  the  pin  at  G.  The  broken  line  COE  is  the  path  of  the 
light  from  C  to  the  eye  when  the  pins  were  seen  apparently  in 
a  straight  line. 

Remove  the  sheet  of  paper  from  the  board,  place  it  over  the 
record  sheet,  and  locate  on  it  the  point  0  and  the  lines  AB, 
C0}  and  OE  by  pricking  pinholes  through  at  two  points  on 
each  line  (where  the  paper  is  not  torn).  The  lines  must  be 
accurately  located;  the  points  where 
the  pins  were  need  not  be.  Draw  the 
lines  AB,  CO,  and  OE  on  the  record 
sheet.  Erect  a  perpendicular  (MN)  to 
AB  at  0.  (For  a  right  angle  to  meas- 
ure by,  fold  a  piece  of  paper  so  that 
the  two  parts  of  a  straight  edge  of  it 
fit  accurately  together.)  Lay  off  on 
OC  and  OE  equal  distances  OF  and 
OG  (not  less  than  5  cm.)  ;  and  from  F 
and  G  drop  perpendiculars  to  MN  (FH  and  GI).  Measure 
FH  and  GI  as  accurately  as  possible  and  compute  GI:FH. 
This  is  the  index  of  refraction  of  water  (Art.  42). 

b.  Repeat  the  experiment  with  the  point  D  in  a  different 
position,  so  that  the  angles  of  incidence  and  refraction  will  be 
different  (larger  if  possible). 

c.  Compute  the  per  cent  of  difference  between  the  two  re- 


166  LIGHT 

suits.     This  difference  is  due  to  experimental  errors  or  to  errors 
of  construction  and  measurement,  and  should  not  exceed  3%. 

Apparatus.  —  Battery  jar;  bit  of  tin  or  other  bright  metal; 
metric  rule. 

Second  Method.  —  Fill  the  jar  nearly  full  of  water  and  drop 
the  bit  of  tin  into  it.  Place  the  tin  (with  the  rule)  so  that 
it  lies  flat  on  the  bottom  and  touches  the  side  of  the  jar. 
Look  with  both  eyes  vertically  down  through  the  water  at  the 
tin  and  through  the  air  at  the  point  of  your  pencil  held  against 
the  outside  of  the  jar.  Move  the  head  slightly  from  side  to 
side,  and,  by  means  of  parallax,  place  the  point  of  the  pencil 
on  a  level  with  the  apparent  position  of  the  tin.  Measure 
(outside  the  jar)  the  distance  from  the  pencil  to  the  top  of  the 
water.  Make  several  trials.  Practice  till  independent  measure- 
ments agree  within  2  or  3  mm.,  and  take  the  average  of  two 
or  three  of  these.  Measure  (inside  the  jar)  the  depth  of  the 
water.  Compute  the  index  of  refraction  of  water  (Art.  43). 

EXERCISE   54.     TOTAL   REFLECTION 

References.  —  Hoadley,  472-473  ;  Carhart  and  Chute,  266- 
268 ;  Slate,  210-211. 

To  study  phenomena  due  to  total  reflection  and  the 
conditions  under  which  they  occur. 

Apparatus.  —  Glass  jar,  preferably  rectangular ;  test  tube ; 
prism. 

NOTE.  —  It  will  be  useful,  both  in  explaining  the  following  experiments 
and  in  drawing  the  figures,  to  know  that  the  critical  angle  for  water  is 
48.5°,  for  crown  glass  about  41°,  and  for  flint  glass  about  37°. 

a.  Stick  a  bit  of  gummed  paper  on  the  outside  of  the  jar  5 
or  6  cm.  from  the  top.  Fill  the  jar  level  full  of  water,  and 
stand  it  near  the  edge  of  the  table  with  the  bit  of  paper  on  the 
side  opposite  you.  Look  at  the  paper  nearly  vertically  through 


TOTAL   REFLECTION 


167 


the  surface  of  the  water,  then  gradually  more  and  more  ob- 
liquely, till  the  eyes  are  on  a  level  with  the  surface  of  the 
water ;  observing  all  the  time  the  simultaneous  change  in  the 
apparent  position  of  the  paper.  Continue  to  lower  the  head 
while  looking  upward  through  the  side  of  the  jar  at  the  under 
side  of  the  surface  of  the 
water.  Presently  an  image 
of  the  paper  will  be  seen  by 
reflection  in  this  surface. 
Draw  a  figure  similar  to  Fig. 
73  and  finish  it,  showing  the 
position  of  the  image  of  the 
paper  for  different  positions 
of  the  eye. 

Why   does    the    image   by 

reflection  not  appear  as  soon 

,  FIG.  73. 

as   the   eyes   are   too   low  to 

see  it  by  refraction  through  the  surface  of  the  water  ? 

b.  With  the  eyes  directed  upward  toward  the  surface  of  the 
water,  observe  in  it  the  image  of  your  pencil,  held  partly  under 
the  water. 

Briefly  describe  its  appearance.  What  evidence  is  there 
that  the  image  is  formed  by  total  reflection  ? 

Try  to  look  through  the  surface  of  the  water  at  the  portion 
of  the  pencil  above  it.  State  and  explain  the  result.  (If  the 
part  of  pencil  above  the  water  is  not  seen,  it  is  not  because  no 
light  enters  the  water  from  it,  but  because  the  light  that  does 
so  does  not  enter  the  eye.) 

c.  Put  a  strip  of  paper  into  the  test  tube ;  and  thrust  the 
tube,  slightly  inclined,  into  the  jar  of  water  with  the  closed 
end  down.     Observe  the  appearance  of  the  portion  of  the  test 
tube  under  water  when  viewed   from   above;   and   note   the 
effect  of  laying  a  sheet  of  white  paper  on  the  table  beside 
the  jar  on  the  side  toward  which  the  lower  end  of  the  tube 
points. 


168 


LIGHT 


FIG.  74. 


From  what  direction  does  the  light  come  that  is  reflected 

into  the  eye  by  the  tube  ? 

Figure  74  represents  a 
section  of  the  test  tube  (the 
thickness  of  the  glass  being 
magnified)  and  its  effect 
upon  several  rays  falling 
upon  it,  one  being  from 
the  paper  inside  the  tube. 
Study  this  figure  for  an 
explanation  of  what  you 
see  when  you  look  at  the 
tube  and  the  paper  inside 
it  from  different  positions. 

Describe  and  explain 
what  you  have  observed,  re- 
ferring to  a  copy  of  Fig.  74. 

d.  Eepeat  with  the  tube  partly  filled  with  water.     Compare 
results  with  the  preceding,  and  explain. 

e.  Lay  a  glass  prism,  face  downward,  on  a  printed  page. 
Look  at  the  page  through  the  prism,  with  the  eyes  at  first 
directly  above  it.     Slowly  lower  the  head  so  as  to  view  the 
page    more    and    more    obliquely 

through  the  near  side  of  the  prism 
until  the  printing  is  no  longer 
visible. 

Describe  the  appearance  of  the 
lower  face  of  the  prism  and  ex- 
plain the  disappearance  of  the 
printing,  referring  to  a  copy  of 
Fig.  75. 

/.  With  the  eyes  in  such  a  posi- 
tion that  the  printing  is  invisible, 
test  the  reflecting  power  of  the 
lower  face  of  the  prism  by  viewing  in  it  the  image  of  a  finger, 


FIG.  75. 


THE    CONVEX   LENS  169 

held  near  the  farther  face  of  the  prism.  While  still  viewing 
this  image,  slowly  raise  the  head  till  the  printing  becomes 
visible,  and  note  the  change  in  the  brightness  of  the  image. 
Give  a  brief  account  of  what  you  have  observed,  and  explain. 

EXERCISE  55.  THE  CONVEX  LENS 

References. — Hoadley,  477-480;  Carhart  and  Chute,  269- 
274;  Jones,  Light,  60-64,  68,  and  72-74. 

Apparatus.  —  Optical  bench  or  meter  rod ;  lens ;  two  card- 
board screens,  one  having  a  circular  hole  about  5  cm.  in  diameter 
at  the  same  height  as  the  lens ;  mounted  candle. 

I.  To  find  the  focal  length  of  a  convex  lens;  and  to 
study  real  images  formed  by  it. 

a.  Turn  the  optical   bench  (or  rod)  toward  some  distant 
object  seen  through  an  adjacent  window.     (A  more  distinct 
image  is  obtained  if  the  window  is  open.)     Place  the  lens  about 
the  middle  of  the  bench,  and  place  near  it,  on  the  side  opposite 
the  object,  the  screen  with  the  hole.     Place  the  other  screen 
behind  this,  and  adjust  it  so  that  a  distinct  image  of  the  distant 
object  is  formed  upon  it.     The  screen  with  the  hole  is  to  inter- 
cept as  much  of  the  light  from  other  sources  as  possible.     Try 
the  effect  of  removing  it. 

Measure  the  distance  from  the  lens  to  the  image.  (If  the 
lens  and  screen  are  mounted  at  the  middle  of  blocks  of  the 
same  length,  measure  from  an  end  of  one  block  to  the  corre- 
sponding end  of  the  other.)  This  is  the  focal  length  (/)  of 
the  lens.  Make  three  settings  of  the  screen  and  average  the 
readings. 

b.  Kemove  the  screen  on  which  the  image  was  focused,  and 
stand  the  screen  with  the  hole  in  its  place,  so  that  the  image  is 
now  in  the  air  in  the  hole  of  this  screen.     Stand  a  meter  or 
more  from  the  lens  on  the  same  side  as  the  screen,  and  view 
the  image  in  the  hole  directly  with  both  eyes.     The  eyes  must 


170  LIGHT 

be  in  line  with  the  hole  in  the  screen  and  the  lens.  The  screen 
serves  merely  to  locate  the  image,  thus  helping  the  observer  to 
direct  his  eyes  toward  its  real  position.  Without  the  screen, 
the  observer  naturally  looks  at  the  lens ;  in  which  case  the 
image  seems  to  be  in  the  lens  and  appears  blurred,  just  as  your 
finger  does  if  you  hold  it  up  before  you  and  look  at  some  object 
a  foot  or  two  beyond  it.  Remove  the  screen  and  try  to  see  the 
image  where  it  really  is.  When  the  eyes  are  properly  directed, 
the  image  appears  perfectly  distinct  and  definitely  located  at 
the  principal  focus. 

Try  viewing  the  image  directly  from  different  positions. 
Can  you  see  it  from  as  many  different  directions  as  you  can 
when  it  is  caught  upon  a  screen  ?  Explain. 

II.  To  study  tJw  relation  between  the  position  of  the 
object  and  the  size,  and  position  of  its  real  image. 

a.  Draw  the  blinds  so  as  to  darken  the  room  (or  carry  the 
apparatus  to  a  darker  room),  turn  the  bench  lengthwise  with  the 
table,  place  the  lens  near  the  middle  of  the  bench,  and  set 
the  lighted  candle  near  one  end.     Stand  a  short  distance  beyond 
the  opposite  end  of  the  bench,  and  look  for  the  image  of  the 
candle  in  front  of  the  lens.     If  you  have  difficulty  in  focusing 
the  eyes  properly,  use  the  screen  to  locate  the  image. 

Let  one  student  slowly  move  the  candle  from  the  lens,  while 
the  other  observes  the  simultaneous  change  in  the  size  and 
position  of  the  image.  Eecord  the  changes  observed. 

b.  Continue  these  observations  till  the  candle  is  so  far  away 
that  the  image  almost  ceases  to  be  visible. 

Does  the  image  continue  to  change  in  size  and  position  as  it 
did  at  first  ?  Does  it  change  as  rapidly  ? 

Which  moves  the  faster,  the  object  or  the  image  ? 

What  is  the  least  distance  of  the  image  from  the  lens  ?  How 
does  this  distance  compare  with  the  focal  length  of  the  lens  ? 

c.  Now  move  the  candle  slowly  toward  the  lens  and  follow 
the  behavior  of  the  image.     To  study  a  large  arid  distant  image 


THE    CONVEX    LENS 


171 


it  is  necessary  to  catch  it  on  the  screen.  Record  the  observed 
changes  in  the  size  and  position  of  the  image. 

Which  moves  the  faster,  the  object  or  the  image  when  the 
object  is  near  the  lens  ? 

When  you  have  a  distinct  image  as  far  from  the  lens  as  you 
can  get  it,  how  far  is  the  candle  from  the  lens  ? 

d.  What  reciprocal  relations  have  you  discovered  between 
the  image  and  the  object  ? 

44.  Formula  for  Convex  Lenses.  —  For  the  present  purpose 
(that  of  establishing  certain  geometrical  relations),  there  is  no 
appreciable  error  in  drawing  rays  as  if  light  were  refracted  once 
halfway  between  the  surf  aces  "of  a  lens  instead  of  at  each  sur- 


FIG.  76. 

face.  The  rays  are  so  drawn  in  Fig.  76.  Let  p  denote  the 
distance  of  the  object  (CO),  p'  the  distance  of  the  image  (OD), 
and  /  the  focal  length  of  the  lens  (OF).  From  similar  triangles 
prove  that  AB:ab  =  CO:OD  or  p:p'-,  also  that  MN:ab  = 
OFiFDoTfi(p'-f). 
Since  AB  =  MN9  we  have  from  these  proportions 

AB :  ab  =  MN:  ab  =p:p'  =/:  (p'  -/). 
From  the  last  proportion  derive  the  formula 

1  +  1=1. 
p  P'  f 

Expressed  in  words,  this  means  that  the  sum  of  the  recipro- 
cals of  a  pair  of  conjugate  focal  distances  of  a  convex  lens  is 
equal  to  the  reciprocal  of  its  focal  length. 


172 


LIGHT 


EXERCISE   56.     THE   FOCAL   LENGTH   OF  A   LENS 

References.  —  The  same  as  for  the  preceding  exercise. 

To  find  the -local  length  of  a  lens  from  its  relation  to 
conjugate  focal  distances. 

Apparatus.  —  Optical  bench  or  meter  rod ;  two  screens,  one 
with  small  hole  and  cross  wires;  flat  gas  jet  or  lamp;  lens. 

a.  Find  the  focal  length  (/)  of  the  lens  by  focusing  it  on  a 
distant  object,  as  in  the  preceding  exercise. 

b.  Draw  the  blinds  so  as  to  make  the  room  rather  dark. 
Stand  the  burner  at  the  end  of  the  optical  bench,  light  it,  and 
turn  the  flame  flatwise  toward  the  bench.     Set  the  screen  with 
the  cross  wires  on  the  end  of  the  bench  near  the  light  (Fig.  77). 
Place  the  lens  at  a  distance  from  the  cross  wires  equal  to  twice 


FIG.  77 


its  focal  length.  Place  the  other  screen  so  that  the  image  of 
the  cross  wires  is  sharply  focused  upon  it.  The  image  of  the 
wires  may  be  red,  black,  or  greenish  blue,  depending  upon 
the  position  of  the  screen.  The  adjustment  should  be  such 
that  the  image  is  as  nearly  black  as  possible.  Let  p  denote  the 
distance  of  the  lens  from  the  cross  wires,  andp'  the  distance  of 
the  image  from  the  lens.  Eecord  as  indicated  below. 

c.   Move  the  screen  on  which  the  image  is  caught  30  to  40  cm. 
farther  from  the  cross  wires ;  then  move  the  lens  toward  the 


THE  FOCAL  LENGTH  OF  A  LENS 


173 


cross  wires  till  the  image  is  again  distinct  upon  the  screen. 
Measure  p  and  p'. 

d.  Without  changing  the  position  of  either  screen,  move  the 
lens  away  from  the  cross  wires  till  the  image  is  again  distinct 
upon  the  other  screen.  Measure  p  andp'. 

Perform  the  computations  indicated  in  the  form  of  record. 
This  will  show  how  nearly  your  results  agree  with  the  formula 
derived  in  Art.  44. 

FOKM  OF  EECORD 


ERROR  =- 

\ 

1 

1      1 

1 

p 

!>' 

P 

_i+-L 
p     p? 

-(Ul) 

v  y 

c       











d 









— 

Discussion.  —  a.    Why  should  p  and  p'  be  equal  in  the  first 
set  of  measurements?     (Find  the  answer  from  the  formula.) 

b.  Why  should  p1  and  p  in  the  last  set  of  measurements  be 
respectively  equal  to  p  and  p'  of  the  preceding  set  ? 

c.  Average  the   numbers  in  the  column  headed  -  H — ,  and 

p     p1 

compute   the   reciprocal  of   this  average.     The   result  is  the 
focal  length  of  the  lens  found  by  the  method  of  conjugate  foci. 

d.  Prove  geometrically  from  a  drawing  like  Fig.  76  that 

Length  of  image  :  length  of  object  =  p' :  p. 

e.  From  a  study  of  the  drawing  discover  whether,  for  a  given 
object  at  a  given  distance,  the  size  and  distance  of  the  image 
would  be  greater  or  less  if  a  lens  of  greater  focal  length  were 
used.     Prove  that,  under  these  conditions,  the  size  of  the  image 
is  proportional  to  its  distance  from  the  lens. 


174 


LIGHT 


45.  Angular  Size  and  Visual  Angle. — It  is  a  familiar  fact 
that  the  distinctness  with  which  an  object  is  seen  depends 
upon  its  distance  from  the  observer  as  well  as  upon  its  size. 
The  reason  for  this  is  shown  in  Fig.  78.  The  crystalline  lens 
(aided  by  the  other  refractive  media  of  the  eye)  forms  upon  the 
back  part  of  the  eyeball  (the  retina)  a  real,  inverted  image  of 
objects  within  view.  The  nearer  an  object  is  to  the  eye  the 
larger  is  its  image  on  the  retina.  In  fact,  the  size  of  the  image 
is  proportional  to  the  angle  under  which  the  object  is  seen 
(angles  AOB  and  A' OB'  in  the  figure)  ;  and  this  angle  is  (very 
nearly)  inversely  proportional  to  the  distance  of  the  object. 


FIG.  78. 


The  angle  under  which  an  object  is  seen  is  called  the  visual 
angle,  and  it  measures  the  angular  size  of  the  object.  The 
distinctness  with  which  an  object  is  seen  (so  long  as  the  image 
is  clearly  defined  upon  the  retina)  is  determined  by  the  visual 
angle ;  and  this,  as  we  have  seen,  varies  inversely  as  the 
distance  of  the  object  for  vision  with  the  naked  eye. 

The  various  forms  of  telescopes  and  microscopes  serve  the 
purpose  of  increasing  the  visual  angle. 

46.  Magnification  of  a  Simple  Microscope.  —  In  using  a  convex 
lens  as  a  simple  microscope,  it  is  commonly  held  close  to  the 
eye;  hence  the  angle  under  which  the  image  is  seen  is  approxi- 
mately a  Ob  (Fig.  79).  It  might  be  supposed  that  the  angular 
size  of  the  image,  and  hence  its  distinctness,  would  be  no 
greater  than  that  of  the  object,  since  angle  aOb  and  angle  AOB 


THE   SIMPLE   MICROSCOPE 


175 


are  identical.  But  in  order  to  view  the  object  directly,  its 
distance  must  be  at  least  as  great  as  EO  (about  25  cm.),  for  at 
a  less  distance  the  eye  is  unable  to  focus  the  image  distinctly 
upon  the  retina  and  the  object  appears  blurred.  At  this  dis- 
tance the  visual  angle  of  the  object  would  be-4'OJB';  hence  the 
magnification  is 

angle  a  Ob :  angle  A' OB'; 
and  this  is  approximately  equal  to 

abiA'B'  orab-.AB. 
From  similar  triangles, 

ab:AB::EO:DO. 


FIG.  79. 


Hence  the  magnification  is  equal  to  the  ratio  of  the  distance 
of  the  image  to  the  distance  of  the  object. 

JEJO,  the  nearest  distance  of  distinct  vision,  is  about  25  cm. 
for  the  normal  eye ;  and,  for  lenses  of  short  focus  (5  cm.  and 
under),  DO  is  but  little  less  than  the  focal  length  of  the  lens; 
hence  the  magnifying  power  of  such  lenses  is  approximately 
expressed  by  the  ratio 

25:/, 

where  /  is  the  focal  length  of  the  lens  in  centimeters. 


17fi  LIGHT 

EXERCISE   57.     THE    SIMPLE   MICROSCOPE 

References.  —  Hoadley,  513  ;  Carhart  and  Chute,  295 ;  Slate, 
216. 

To  study  the  formation  of  virtual  images  by  a  convex, 
lens. 

Apparatus.  —  Optical  bench;  mounted  lens  of  short  focal 
length ;  mounted  candle ;  screen. 

a.  Find  the  focal  length  of  the  lens  by  focusing  on  a  distant 
object. 

b.  Place  the  lighted  candle  on  the  end  of  the  optical  bench 
and  the  lens  in  front  of  it  at  a  distance  considerably  greater 
than  its  focal  length.     Catch  the  image  of  the  candle  on  the 
screen  (or  a  sheet  of  paper).     (It  may  be  necessary  to  go  to 
a  dark  corner  of  the  room  for  this.)     Remember  that  a  real 
image  is  formed  Jby  the  convergence  of  light  from  each  point 
of  the  object  to  the  corresponding  point  of  the  image.     Move 
the  candle  slowly  toward  the  lens,  meanwhile  following  the 
image  with  the  screen. 

As  the  candle  is  moved  up,  does  the  cone  of  light  that 
falls  upon  the  lens  from  any  point  of  it  become  more  or  less 
divergent  ? 

Does  the  refracted  cone  become  more  or  less  convergent  ? 

c.  Move  the  candle  up  to  the  principal  focus  of  the  lens. 
What  has  become  of  the  image  meanwhile  ? 

What  is  now  the  shape  of  the  cone  of  light  from  a  point  of 
the  candle  after  passing  through  the  lens  ? 

d.  With  the  eye  close  to  the  lens,  look  through  it  at  the 
candle  while  you  are  moving  it  up  close  to  the  lens.     What 
you  see  is  not  the  candle,  but  its  image. 

On  which  side  of  the  lens  is  it  ? 
Can  it  be  caught  on  a  screen  ?     Is  it  real  or  virtual  ? 
Are   the   rays   from   a  point  of  the  candle  convergent  or 
divergent  after  passing  through  the  lens  ? 

How  does   the   image   compare   in    size   and  position  with 


COLOR  177 

the  object?  Be  careful  as  to  its  position;  the  eye  is  easily 
deceived  on  this  point.  Locate  the  image  by  parallax  by 
holding  a  pencil  just  above  it,  looking  at  the  pencil  over  the 
lens.  Moving  the  head  slightly  from  side  to  side  will  help. 

e.  What  change  takes  place  in  the  size  and  position  of  the 
image  as  the  object  is  brought  up  toward  the  lens  from  the 
principal  focus  ?  Explain  this  change  of  size  and  position 
by  means  of  two  drawings,  one  with  the  distance  of  the  object 
(an  arrow)  only  slightly  less  than  the  focal  length  and  one 
with  the  object  very  near  the  lens. 

/.  Use  the  lens  as  a  simple  microscope  in  viewing  various 
objects,  as  the  back  of  your  hand,  cloth,  printing,  etc.  Hold 
the  lens  close  to  the  eye  and  vary  the  distance  of  the  object 
till  it  is  seen  distinctly.  While  looking  at  the  different  objects 
estimate  the  magnification  of  the  lens.  jMagnification  is  always 
the  ratio  of  corresponding  linear  dimensions  of  image  and  ob- 
ject, never  of  surfaces  or  volumes. 

g.  How  does  the  estimated  magnification  compare  with  the 
value  given  by  the  formula  25  -s-/? 


EXERCISE   58.     COLOE 

References.  —  Hoadley,  486-487  and  495-499;  Carhart  and 
Chute,  277-278  and  287-292 ;  Slate,  218-221 ;  Jones,  Light,  97. 

Apparatus.  —  Glass  prism  ;  small  square  of  black  cardboard 
with  a  slit  1  mm.  by  2  cm.  and  a  slit  1  cm.  by  2  cm.;  strips 
of  colored  paper  1  mm.  wide  on  black  cardboard ;  small  squares 
or  other  patterns  of  complementary  colors  pasted  on  opposite 
sides  of  black  cardboard  (yellow  and  blue  on  one  and  purple 
and  green  on  another) ;  pieces  of  colored  glass  (blue  and 
yellow) ;  colored  disks  and  top. 

I.  To  separate  li$ht  into  its  elementary,  or  prismatic, 
colors. 

a.  Stand  facing  a  window  and  hold  the  cardboard  with  the 
slits  at  about  the  height  of  the  eyes,  with  the  slits  horizontal 
COLEMAN'S  PHY.  LAB.  MAN.  — 12 


178 


LIGHT 


FIG.  80. 


and  strongly  illuminated  by  sunlight  (having  the  sky  for  a 
background).     Look  at  the  narrow  slit  through  the  prism,  held 

as  shown  in  the  figure, 
with  its  edges  parallel  to 
the  slit.  The  prism  should 
be  held  near  the  eye,  and 
the  slit  at  a  distance  of 
about  a  foot.  Make  a 
drawing  like  Fig.  80,  but  larger,  and  complete  it,  showing  the 
apparent  position  and  the  appearance  of  the  slit.  The  drawing 
should  account  for  the  relative  position  of  the  red  and  blue 
ends  of  the  spectrum  a§.  it  appears  upon  the  cardboard. 

b.  Look  at  the  wide  slit  in  the  same  way.     Account  for  the 
absence  of  color  from  the  central  portion  of  the  slit,  and  draw 
a  figure  to  illustrate.  ^ 

c.  Cover  one  end  of  the  narrow  slit  with  the  blue  glass,  and 
view  the  slit  through  the  prism  as  before.     Compare  the  spec- 
tra of  the  covered  and  the  uncovered  portions  of  the  slit. 

What  colors  are  transmitted  through  the  glass  ?  What 
colors  are  absorbed  by  it  ? 

Why  does  the  glass  appear  blue  when  it  transmits  other 
colors  also  ? 

d.  Analyze  in  the  same  way  the  light  transmitted  by  the 
other  pieces  of  glass. 

e.  Hold  the  colored  strips  on  the  black  cardboard  so  that 
they  are  illuminated  by  direct  sunlight,  and  with  the  prism 
analyze  the  light  reflected  by  them.     Record  as  follows,  using 
the  initial  letters  V,  I,  B,  Gr,  Y,  0,  and  E  to  denote  the  colors:  — 


COLOR  OF 
STRIP 

Col.OKS    COMPOSING 

REFLECTED  LIGHT 

('oU)RS    AP.SOKBKI) 

BY  STRIP 

white 
red 

etc, 

COLOR  179 

II.  To  observe  the  colors  produced  when  different  colors 
are  combined  by  rapid  rotation. 

a.  Fasten  a  red  and  a  yellow  disk  to  the  top,  leaving  half 
of  each  exposed.  Spin  the  top,  and  observe  the  resultant 
color.  Try  in  the  same  way  the  red  and  violet  and  the  blue 
and  green  disks.  If  you  try  other  combinations,  make  a  record 
of  them,  also.  Note  the  relative  positions  of  the  colors  used 
and  of  the  resultant  color  in  the  chart  of  complementary  colors 
given  in  the  text  or  reference  books. 

What  is  the  position  on  the  chart  of  the  resultant  color, 
relative  to  the  colors  combined  to  produce  it  ? 


COMPONENT  COLORS 

KESULTANT  COLOK 

red  and  yellow 
red  and  violet 
blue  and  green 

6.  When  united  in  this  way,  complementary  colors,  if  suffi- 
ciently intense,  will  produce  white.  Generally,  however,  the 
whole  amount  of  light  reflected  is  so  small  that  the  resulting 
color  is  gray,  or  even  as  dark  as  slate.  That  such  colors  are 
really  shades  of  white  (that  is,  white  of  a  low  degree  of  lumi- 
nosity) may  be  shown  by  rotating  together  the  white  and  the 
black  disks,  first  with  but  little  of  the  black  exposed,  then  with 
more  and  more  of  the  black.  Test  the  white  and  black  in  this 
way. 

Try  the  different  pairs  of  complementary  colors  provided. 
The  yellow  and  blue  disks  will  probably  give  the  best  result. 
If  in  any  case  the  resulting  color  is  not  a  neutral  gray  or  slate, 
vary  the  portion  of  each  color.  Draw  circles  and  divide  them  so 
as  to  show  the  portion  of  each  color  when  the  resultant  is  gray. 

c.  Try  to  produce  white  (gray)  with  a  combination  of  the 
primary  colors,  red,  green,  and  violet.  Make  a  drawing  as 
before. 


180  LIGHT 

III.  To  observe  after  images-;  and  to  -find  whether  a 
color  and  its  after  image  are  complementary. 

a.  Look  steadily,  without  moving  the  eyes,  for  about  half  a 
minute  at  the  green  paper  on  the  black  cardboard,  with  as 
strong  a  light  on  the  paper  as  you  can  get  (direct  sunlight  if 
possible) ;  then  quickly  look  at  a  sheet  of  white  paper. 

What  is  the  shape  and  color  of  the  spot  observed  ? 

Does  the  spot  follow  the  eye  as  you  look  at  different  parts  of 
the  paper  ?  The  spot  is  called  an  after  image. 

(Consult  the  text  or  a  reference  book  for  the  cause  of  after 
images.) 

After  again  looking  steadily  at  the  paper  for  half  a  minute, 
close  the  eyes  quickly  and  hold  the  hand  over  them  to  exclude 
the  light  that  would  otherwise  penetrate  the  eyelids. 

What  is  the  shape  and  color  of  the  spot  observed  ? 

b.  Repeat  the  'experiment  with  the  bit  of  purple  paper.     (If 
purple  is  not  provided,  use  violet.) 

What  relation  does  the  color  of  the  after  image  bear  to  that 
of  the  object  ? 

c.  Try  the  yellow  and  the  blue  papers,  observing  the  after 
images  with  the  eyes  closed  or  by  looking  at  white  paper. 

EXERCISE   59.     SPECTRA 

References.  —  Hoadley,  500-506;  Carhart  and  Chute,  283- 
286;  Sanford,  pp.  392-398. 

Apparatus.  —  Spectroscope ;  Bunsen  burner ;  flat  gas  jet ; 
ring  stand ;  asbestos ;  blue,  yellow,  and  red  glass ;  solutions 
of  strontium,  barium,  calcium  and  potassium  salts ;  platinum 
wire  for  each  solution ;  platinum  foil ;  black  screen. 

[Make  a  hole  about  2  cm.  in  diameter  in  the  middle  of  a  small  piece  of 
sheet  asbestos,  and  rub  salt  into  the  asbestos  about  the  hole.  A  Bunsen 
flame  rising  through  this  hole  will  be  strongly  colored  with  sodium.  Fuse 
pieces  of  platinum  wire  7  or  8  cm.  long  and  the  platinum  foil  into  short 
pieces  of  glass  tubing  for  handles.  Roll  2  cm.  of  the  wire  at  the  end  into 
a  small,  tight  coil.] 


SPECTRA  181 

I.    To  study  and  adjust  a  spectroscope. 

a.  If  the  prism  is  covered,  remove  the  cover  and  observe  the 
position  of  the  prism.     Find  from  its  position  what  the  direc- 
tion of  deviation  must  be,  and  in  what  position  the  telescope 
must  be  placed  to  receive  the  light  from  the  collimator  after  it 
has  passed  through  the  prism. 

The  telescope  must  be  focused  for  parallel  rays ;  hence  the 
eyepiece  is  to  be  adjusted  so  that  distant  objects  (viewed 
through  the  open  window)  are  seen  distinctly.  With  some 
instruments  an  unobstructed  view  of  distant  objects  can  be 
obtained  by  removing  the  prism  (do  not  touch  the  refracting 
surfaces) ;  with  others  it  will  be  necessary  to  unscrew  the  tele- 
scope. If  the  telescope  is  removed,  be  very  careful  in  replacing 
it  not  to  damage  the  thread  by  starting  it  the  wrong  way  and 
forcing  it.  This  adjustment  of  the  telescope  for  distant  ob- 
jects is  to  remain  unchanged  throughout  the  exercise. 

b.  If  the  telescope   is  in  a  fixed  position,  omit  this  para- 
graph.    If  it  is  carried  on  a  movable  arm,  turn  it  into  line 
with  the  collimator ;  and,  with  the  prism  removed,  move  the 
adjustable  end  of  the  collimator  so  that  the  slit  is  seen  dis- 
tinctly through  the  telescope. 

c.  Light  the  Bunsen  burner,  and  place  it  in  line  with  the 
collimator  about  10  cm.  beyond  its  end.     Support  the  asbestos 
on  the  ring  stand,  and  adjust  it  so  that  it  is  a  little  below  the 
level  of  the  collimator  and  so  the  flame  of  the  burner  rises 
through  the  hole  in  the  asbestos.     If  the  flame  is  not  strongly 
colored  yellow,  rub  more  salt  into  the  asbestos  about  the  hole. 
The  yellow  color  is  due  to  sodium  vapor.     Place  the  black 
screen  a  short  distance  beyond  the  burner,  so  as  to  shut  out 
other  light  from  the  collimator.     Open  the  slit  of  the  collimator 
a  little  way  by  turning  the  thumbscrew  at  the  side,  and  look 
through  the  telescope.     If  the  apparatus  is  properly  adjusted, 
you  will  see  a  yellow  image  of  the  slit.     If  this  image  is  not 
distinct,  move  the  adjustable  end  of  the  collimator  in  or  out 
till  it  is.     The  slit  should  be  vertical.     This  adjustment  of  the 


iXiJ  LIGHT 

distance  of  the  slit  is  to  remain  unchanged  throughout  the 
exercise;  but  the  width  of  the  slit  may  be  changed  as  desired. 
Observe  the  effect  of  varying  the  width  of  the  slit,  and  leave  it 
finally  about  the  width  of  a  fine  hair. 

cl  This  paragraph  is  to  be  included  or  omitted  with  para- 
graph b.  If  b  is  omitted,  c  is  done  with  the  prism  in  place. 
If  b  is  included,  replace  the  prism  now,  and  cover  it  and  the 
adjacent  ends  of  the  telescope  and  collimator  so  as  to  exclude 
as  much  light  as  possible.  Turn  the  telescope  in  the  proper 
direction  till  the  image  of  the  slit  is  seen. 

Has  the  prism  affected  its  size  or  color  ?  What  is  the  effect 
of  the  prism  ? 

e.  If  the  spectroscope  has  a  third  tube  carrying  a  scale, 
light  the  gas  tip  in  front  of  it.  Look  in  the  telescope,  and 
adjust  the  sliding  piece  at  the  end  of  the  third  tube  till  the 
scale  that  it  carries  is  seen  distinctly.  If  this  scale  is  not 
horizontal,  turn  the  end  piece  till  it  is.  Again  remove  the 
cover  of  the  prism,  and  observe  that  the  scale  is  seen  by  reflec- 
tion from  the  front  surface  of  the  prism. 

II.    To  study  continuous  spectra. 

a.  Observe   the    spectrum    of   the   Bunsen   flame  with   the 
asbestos  removed.     The  screen  should  be  in  position  to  shut 
out  other  light.     If  the  flame  is  nearly  non-luminous,  as  it 
should  be,  it  will  give  no  spectrum,  or,  at  the  most,  only  a 
very  faint  one.     Hold  a  piece  of  platinum  foil  in  the  flame 
and  in  line  with  the  slit. 

What  kind  of  spectrum  does  it  give  when  incandescent  ? 
What  classes  of  bodies  give  this  kind  of  spectrum  ?  (See 
references.) 

b.  Substitute  the  flat  gas  jet  for  the  Bunsen  burner  and 
observe  its  spectrum. 

Why  is  it  continuous  ?  (Consult  any  chemistry  on  the 
cause  of  the  luminosity  of  a  gas  flame.) 

For  what  color  is  the  angle  of  deviation  greatest  ?     Least  ? 


SPECTRA  183 

III.  To  study  discontinuous  or  bright-line  spectra. 

The  bright-line  spectrum  of  sodium  has  already  been  ob- 
served. Dip  the  wire  for  the  strontium  salt  into  the  solution  of 
that  salt ;  then  hold  it  in  the  Bunsen  flame  and  observe  the  spec- 
trum. Compare  with  the  colored  plate  of  the  spectrum  in 
some  reference  book.  Draw  a  figure,  locating  the  lines  of  the 
spectrum  by  the  scale  of  the  instrument,  if  it  has  one,  and 
name  their  color  in  the  figure. 

Try  the  other  salts,  using  for  each  the  wire  provided  for  it ; 
and  compare  with  colored  plates  of  the  spectra.  In  each  case 
the  spectrum  observed  is  that  of  the  metal  in  the  salt.  Draw 
figures. 

Is  the  metal  in  the  solid  or  gaseous  state  while  emitting  the 
light  ? 

IV.  To  study  absorption  spectra. 

a.  Obtain  a  continuous  spectrum  from  the  flat  gas  jet;  then 
hold  the  blue  glass  between  the  flame  and  the  slit.  What 
colors  are  most  strongly  absorbed  by  the  glass  ?  Which  are 
transmitted  ? 

6.   Test  the  red  and  the  yellow  glass  in  the  same  way. 

c.  What  colors  are  transmitted  by  the  blue  and  the  yellow 
glass  together  ?     Answer  from  the  results  obtained  with  the 
two  separately,  and  test  your  conclusion  by  holding  them  both 
together   before   the    slit   and  observing  the  spectrum  of  the 
transmitted  light. 

Look  through  the  two  together  toward  the  light.  What  is 
the  color  of  the  transmitted  light  ? 

d.  Turn   the   collimator   toward   the    sky,  and  observe  the 
spectrum  of  skylight  (the  solar  spectrum).     If  the  slit  is  narrow 
and  the  telescope  properly  focused,  many  dark  lines  (Fraun- 
hofer  lines)  will  be  seen. 

Where  are  the  substances  that  absorb  the  colors  that  would 
otherwise  occupy  the  places  in  the  spectrum  where  the  Fraun- 
hofer  lines  are  ?  (See  references.) 


184 


LIGHT 


Observe  and  explain  the  effect  upon  these  lines  of  widening 
the  slit. 

e.  If  the  spectroscope  has  a  comparison  prism,  do  the  follow- 
ing ;  if  not,  omit  it.  Narrow  the  slit  till  the  lines  are  distinctly 
visible.  Cover  half  the  slit  with  the  comparison  prism,  and 
stand  the  sodium  flame  at  the  side  so  that  its  light  is  reflected 
by  the  prism  into  the  slit.  You  now  have  the  solar  spectrum 
with  the  spectrum  of  sodium  beside  it.  Observe  whether  the 
sodium  line  is  continuous  with  a  dark  line  in  the  solar  spectrum. 
Explain. 

47.  The  Astronomical  Telescope.  —  In  Fig.  81  MO  and  HI 
represent  rays  from  the  top  of  a  distant  object,  and  NO  and  JK 
rays  from  the  bottom.  The  inverted  arrow  ab  represents  the 
real,  inverted  image  formed  by  the  objective.  Since  the  object 
is  at  a  considerable  distance,  the  distance  of  the  image  (0(7)  is 
the  focal  length  of  the  objective  (F).  The  eye  lens  is  used  as  a 


FIG.  81. 

simple  microscope  to  form  a  magnified,  virtual  image  (a'b1)  of 
the  real  image ;  hence  CE  is  approximately  equal  to  the  focal 
length  of  the  eye  lens  (/). 

By  the  use  of  the  eye  lens  the  visual  angle  becomes  a'Eb'  or 
aEb.  The  magnifying  power  of  the  telescope  is  the  ratio  of  the 
visual  angle  when  the  object  is  viewed  through  it  (angle  aEb) 
to  the  visual  angle  with  the  naked  eye  (angle  MON  or  its  equal 


THE   ASTRONOMICAL   TELESCOPE  185 

angle  a  Ob).     But,  for  small  angles,  the  visual  angle  is  (very 
nearly)  inversely  as  the  distance  (Art.  45)  ;  hence 

the  magnifying  power  =  angle  aEb  :  angle  a  Ob 

=  OC:GE  or  F:f  (very  nearly). 

Stated  in  words  this  means  that  the  magnifying  power  of  a 
telescope  is  directly  proportional  to  the  focal  length  of  the 
objective  and  inversely  proportional  to  the  focal  length  of 
the  eyepiece.  For  example,  the  magnifying  power  would  be 
doubled  either  by  doubling  the  focal  length  of  the  objective 
(which  would  double  the  size  of  the  real  image  ab)  or  by  reduc- 
ing the  focal  length  of  the  eyepiece  one  half  (which  would 
double  the  size  of  the  virtual  image  a'b').  Some  elementary 
text-books  make  the  erroneous  statement  that  the  magnification 
is  due  entirely  to  the  eyepiece. 

EXERCISE   60.     THE   ASTRONOMICAL   TELESCOPE 

References. — Hoadley,  515-516;  Carhart  and  Chute,  297; 
Slate,  216-217 ;  Jones,  Light,  87-88. 

To  study,  by  means  of  two  convex  lenses,  the  use  of 
the  objective  and  the  eyepiece  of  an  astronomical  tele- 
scope and  the  relation  between  their  focal  lengths  and 
the  magnifying  power  of  the  instrument. 

Apparatus.  — Optical  bench ;  mounted  screen ;  three  mounted 
convex  lenses,  one  of  long  focal  length  (the  longer  the  better 
up  to  40  cm.),  and  two  of  unequal  short  focal  length. 

a.  Find  the  focal  length  of  each  of  the  lenses  by  focusing 
the  image  of  a  distant  object  on  the  screen.     Let  F  denote  the 
longest  focal  length,  /j  the  next,  and/2  the  shortest. 

b.  Tarn  the  bench  toward  a  distant  building,  and  stand  the 
lens  having  the  shortest  focal  length  (the  eye  lens)  on  the  bench 
at  the  end  farthest  from  the  object.     Use  the  lens  having  the 


18G  LIGHT 

longest  focal  length  as  the  objective.  Place  it  on  the  bench  at 
a  distance  from  the  eye  lens  approximately  equal  to  the  sum 
of  the  focal  lengths  of  the  two  lenses ;  and,  with  the  eye  close 
to  the  eye  lens,  adjust  the  objective  till  the  image  is  distinct. 
It  will  be  better  to  raise  the  window,  as  the  wavy  surface  of 
common  window  glass  interferes  seriously  with  the  formation 
of  distinct  images. 

View  the  object  directly  with  one  eye,  and  the  image,  at  the 
same  time,  with  the  other;  and  estimate  the  magnification 
(the  ratio  of  the  apparent  length  of  any  part  of  the  image  to 
the  length  of  the  corresponding  part  of  the  object).  The  best 
object  for  this  purpose  is  one  bounded  by  short,  straight  lines, 
as  a  window. 

c.  Compute  the  magnification  from  the  focal  lengths  (Art. 
47),  and  compare  with  the  estimated  value. 

d.  Measure  the  distance  between  the  lenses,  and  compare 
this  distance  with  the  sum  of  the  focal  lengths  of  the  lenses. 

Why  should  these  quantities  be  equal  ?     (See  Art.  47.) 

e.  Why  is  the  image  bordered  with  red  and  blue  ? 

Is  the  coloring  greater  or  less  for  parts  of  the  image  seen 
through  the  central  portion  of  the  eye  lens  ?  Why  ? 

/.  Repeat  the  experiment  with  the  same  objective  and  the 
other  lens  for  the  eye  lens.  Estimate  and  compute  the  magni- 
fication as  before ;  but  omit  paragraph  d. 

g.    Is  chromatic  aberration  greater  or  less  than  before  ?  Why  ? 

h.  Repeat  the  experiment  with  the  lens  of  medium  focal 
length  (/j)  for  the  objective  and  the  one  of  shortest  focal 
length  for  the  eye  lens. 

How  does  the  magnification  compare  with  that  of  the  first 
combination,  where  the  same  eye  lens  was  used  ? 

What  is  the  advantage  of  an  objective  of  long  focal  length? 

Discussion.  —  Copy  Fig.  81,  and  prove,  referring  to  the  figure, 
that  the  size  of  the  image  formed  by  the  objective  is  proportional 
to  its  focal  length. 


THE    GALILEAN   TELESCOPE  187 

EXEKC1SE   61.     THE   GALILEAN   TELESCOPE 

References.  —  Hoadley,  482  and  517 ;  Carhart  and  Chute, 
271-272  and  298;  Jones,  Light,  90. 

Apparatus.  —  Optical  bench ;  convex  lens  of  long  focal  length ; 
concave  lens. 

I.  To  study  the  effect  of  a  concave  lens  on  parallel 
and  diverging  light. 

a.  Try  to  focus  a  beam  of  sunlight  with  a  concave  lens  as 
you  did  with  a  convex  lens.     State  and  explain  the  result. 

b.  Look  at  near  and  distant  objects  through  the  lens,  noting 
the  effect  of  the  lens  on  the  apparent  size  and  position  of  the 
object. 

State  this  effect  and  explain  it  by  a  figure  showing  the  for- 
mation of  an  image  by  a  convex  lens. 

c.  It  is  evident  that,  since  a  concave  lens  makes  parallel 
rays  divergent,  it  will  make  convergent  rays  less  convergent, 
and  may  even  make  them  parallel  or  divergent.     In  the  Gali- 
lean telescope  its  function  is  to  intercept  the  converging  rays 
from  the  objective  and  make  them  very  slightly  divergent 
(practically  parallel). 

II.  To  adjust  and  use  a  convex  and  a  concave  lens  as 
a  Galilean  telescope. 

a.  The  focal  length  of  a  concave  lens  can  be  found  experi- 
mentally, but  not  so  simply  as  that  of  a  convex  lens.  On  this 
account  measurements  and  computations  are  omitted  from  this 
exercise. 

Turn  the  optical  bench  toward  a  distant  building,  and  place 
the  convex  lens  (the  objective)  at  the  nearer  end.  Find  ap- 
proximately the  position  of  the  real  image  formed  by  the  objec- 
tive, and  place  the  concave  lens  (the  eye  lens)  on  the  bench  at 
about  this  point.  While  looking  through  the  eye  lens,  move 
it  slowly  toward  the  objective  till  the  image  is  distinctly  seen. 


188 


LIGHT 


b.  Direct  the  telescope  toward  a  window  of  the  distant 
building  so  that  it  can  be  seen  directly  with  one  eye  and 
through  the  telescope  with  the  other,  and  estimate  the  magni- 
fication as  you  did  for  the  astronomical  telescope. 


FIG.  82. 

Discussion.  —  Copy  Fig.  82  and  write  a  brief  explanation, 
answering  the  following  questions:  — 

a.  Which  rays  come  from  the  top  of  the  object  and  which 
from  the  bottom  ? 

6.    What  (in  the  figure)  is  the  focal  length  of  the  objective  ? 

c.  What  is  the  focal  length  of  the  eye  lens  ? 

d.  Wrhat  is  the  visual  angle  with  the  naked  eye  ? 

e.  What  is  the  visual  angle  with  the  telescope? 

/.    What  ratio  (of  lengths)  measures  the  magnification  ? 


EXERCISE   62.      THE   COMPOUND   MICROSCOPE 

References. — Hoadley,  514;  Carhart  and  Chute,  296;  Jones, 
Light,  89. 

To  study,  by  means  of  two  convex  lenses,  the  use  of  the 
objective  and  the  eyepiece  of  a  compound  microscope;  (in (I 
to  compute  the  magnifying  power. 

Apparatus.  —  Optical  bench ;  two  lenses  of  short  focal  length, 
preferably,  of  the  same  size  and  focal  length ;  printed  page, 
inverted  and  fastened  to  a  mounted  screen. 


THE    COMPOUND   MICROSCOPE 


189 


a.  Find  the  focal  length  of  the  lenses  by  focusing  the  image 
of  a  distant  object  on  the  screen.     Let  F  denote  the  shorter 
focal  length  (if  they  differ)  and /the  longer. 

b.  Place  the  lens  of  longer  focal  length  (/)  at  one  end  of 
the  bench  for  the  eye  lens,  and  the  object  (the  inverted  printed 
page)  at  the  other.     Starting  with  the  other  lens  near  the 


FIG.  83. 

object,  slowly  move  it  toward  the  eye  lens  till  a  distinct  image 
of  one  or  more  letters  is  seen. 

c.  Measure  the  distance  from  the  object  to  the  objective  (p) 
and  the  distance  between  the  lenses  (M). 

Let  pf  denote  the  distance  between  the  objective  and  the  real 
image  formed  by  it  ;  then  M  should  be  very  nearly  equal  to 
/  +  />'.  Why? 

To  test  this  relation,  compute  p'  from  the  formula 


. 
p     p'     F 

To  p'  add  the  focal  length  of  the  eye  lens  (/),  and  compare  the 

sum  with  M. 

p 
d.   The  magnification  by  the  objective  is  —  .     Why? 


190  LIGHT 

95 
The  magnification  by  the  eye  lens  is  ~-     Why  ? 

/r>'\       /^'i 

The  total  magnification  is  (  *L  )  x  (  ^ 


Compute  the  magnification  from  the  measurements  taken. 

e.  Move  the  eye  lens  a  few  centimeters  nearer  the  object, 
then  move  the  objective  toward  the  eye  lens  till  the  image  is 
again  distinct. 

Is  the  magnification  more  or  less  than  it  was  before  ?  Ex- 
plain. 

/.  Eeplace  the  eye  lens  exactly  in  its  first  position;  and 
move  the  objective  toward  the  eye  lens  till  a  second  position 
is  found  for  which  the  image  is  distinct. 

Is  the  magnification  more  or  less  than  in  the  first  instance  ? 
Explain. 

g.  With  this  adjustment  you  should  find  that  the  distances 
p  and  p'  of  paragraph  c  are  interchanged,  the  distance  between 
the  object  and  the  objective  being  p'  and  the  distance  between 
the  lenses/-!-  p.  Why  should  this  be  so  ? 

Measure  the  distance  between  the  object  and  the  objective 
and  compare  it  with  p'. 

h.  The  lenses  in  this  adjustment  make  an  astronomical  tele- 
scope for  viewing  the  printing.  In  what  respects  does  it  differ 
from  the  microscope  ? 


VIII.     MAGNETISM   AND   ELECTRICITY 
EXERCISE   63.     MAGNETS   AND   MAGNETIC  ACTION 

References.  —  Hoadley,  291-298 ;  Carhart  and  Chute,  358-366 
and  377. 

Apparatus.  —  Bar  magnet ;  small  pieces  of  iron,  brass,  copper, 
zinc,  lead,  wood,  glass,  paper,  etc. ;  small  pieces  (about  8  or  10 
cm.  square)  of  cardboard,  glass,  thin  wood,  sheet  iron  (or  tin), 
zinc,  brass,  and  lead ;  coarse  iron  filings  or  small  tacks ;  mag- 
netic needle,  mounted  or  suspended  (a  magnetized  sewing, 
darning,  or  knitting  needle  will  serve). 

I.  To  observe  the  distribution  of  attracting  power  in  a 
magnet. 

Lay  the  bar  magnet  in  a  box  of  coarse  iron  filings  (or  small 
tacks),  so  that  its  whole  length  comes  in  contact  with  the  fil- 
ings. Lift  the  magnet  and  observe  the  distribution  of  the 
filings  that  cling  to  it. 

What  does  the  experiment  show  concerning  the  distribution 
of  magnetic  attraction  in  a  bar  magnet?  The  regions  of 
greatest  attraction  are  called  poles.  Remove  the  filings  by 
wiping  them  toward  the  middle  of  the  magnet. 

II.  To  find  which  of  certain  substances  are  attracted 
by  a  magnet  and  which  are  not. 

Try  to  lift  with  the  magnet  small  bits  of  various  substances  ; 
and  classify  them  as  magnetic  and  non-magnetic  according  to 
whether  they  are  or  are  not  attracted  by  the  magnet. 

191 


192  MAGNKTISM    AM)    KLiiCTKUTFY 

III.  To  find   whether  tlierc    /A  < it  traction  or  repulsion 
between  like  poles;  between  unlike  pole*. 

The  end  of  the  bar  magnet  marked  N  is  the  north-seeking  or 
north  pole.  It  is  like  the  pole  of  the  magnetic  needle  that 
points  toward  the  north.  (This  may  be  tested  by  suspending 
the  bar  magnet  by  a  thread  at  the  middle,  and  observing  the 
direction  in  which  the  marked  end  points  when  it  comes  to 
rest.)  Observe  the  effect  of  bringing  each  of  the  poles  of  the 
bar  magnet  near  each  of  the  poles  of  the  magnetic  needle. 

What  action  is  observed  between  like  poles?  Between  un- 
like poles  ? 

IV.  To  find  which  of  certain  substances  act  as  screens 
to  cut  off  magnetic  action,  and  which  do  not. 

a.  Put  a  small  quantity  of  iron  filings  (or  tacks)  on  the  card- 
board, and  move"  a  pole  of  the  magnet  to  and  fro  against  the 
under  side  of  the  cardboard  beneath  the  filings.    What  evidence 
is  there  of  magnetic  action  through  the  cardboard  ?     Test  in  the 
same  way  the  different  substances  provided,  and  classify  them 
in  two  groups  according  as  they  do  or  do  not  act  as  a  screen  to 
cut  off  magnetic  action.     Substances  that  act  as  a  screen  are 
said  to  be  very  permeable,  for  reasons  that  will  appear  in  the 
next  exercise. 

Gather  up  with  the  magnet  any  scattered  filings. 

b.  Compare  these  lists  with  your  lists  of  magnetic  and  non- 
magnetic substances. 

EXERCISE   64.     MAGNETIC   INDUCTION.     THEORY 
OF   MAGNETISM 

References.— Hoadley,  307-309 ;  Carhart  and  Chute,  367-373 ; 
Slate,  224. 

Apparatus.  —  Bar  magnet ;  magnetic  needle;  piece  of  soft  iron 
rod  (Norway  iron);  piece  of  wood  of  same  size  and  shape  as  the 


MAGNETIC    INDUCTION.     THEORY   OF    MAGNETISM     198 

iron  rod;  piece  of  steel;  iron  wire  10  in.  long  with  an  inch  at 
each  end  bent  at  right  angles ;  iron  filings ;  slender  test  tube 
nearly  full  of  iron  filings. 

[A  satisfactory  magnetic  needle  for  this  and  the  following  exercise  can 
be  made  by  magnetizing  a  piece  of  a  large  sewing  or  darning  needle  an 
inch  long  and  suspending  it  by  a  silk  thread  or,  better,  by  a  single  strand. 
Waxing  the  thread  at  the  end  before  tying  it  to  the  needle  will  keep  it 
from  slipping.  It  will  be  still  better  if  the  thread  is  run  through  a  piece 
of  small  glass  tubing  5  in.  long  and  fastened  with  wax  at  the  farther  end 
so  as  to  permit  the  needle  to  hang  freely  a  little  below  the  tube  when  it  is 
held  vertically.  The  tube  prevents  lateral  motion  of  the  needle.] 

I.  To  find  what  is  meant  by  magnetic  permeability  • 
and  to  study  magnetic  induction  in  iron  and  steel. 

a.  Try  to  pick  up  iron  filings  with  the  soft  iron  rod.  Does 
it  show  magnetic  properties?  Try  again  with  a  pole  of  the 
magnet  held  against  the  other  end  of  the  rod.  What  happens 
to  the  load  of  filings  on  the  end  of  the  rod  when  the  magnet  is 
removed  from  the  other  end  ? 

6.    Test  the  piece  of  wood  as  you  did  the  iron  rod. 

c.  The  magnetic  action  permeates  the  iron,  extending  from 
end  to  end,  the  rod  serving  as  a  carrier  for  the  action.     This  is 
what   is   meant   by   calling    magnetic    substances    permeable. 
Compare  the  results  obtained  here  with  those  of  Part  IV  of 
the  preceding  exercise. 

d.  Hold  a  pole  of  the  magnet  against  an  end  of  the  soft  iron 
rod,  and  test  the  polarity  of  the  magnetism  induced  in  the  iron 
by  bringing  the  other  end  of   it   near   the   magnetic   needle. 
Repulsion  rather  than   attraction    should   be    made   the   test. 
\Vliy?     Repeat  with  the  other  end  of  the  magnet  held  to  the 
iron. 

If  a  more  complete  test  of  the  magnetism  induced  in  the 
iron  were  made,  it  would  be  found  to  have  unlike  poles  at  its 
ends.  Assuming  this  to  be  true,  is  the  pole  at  the  end  touched 
by  the  magnet  like  or  unlike  the  inducing  pole  ? 

COLKMAN'S  PHY.  LAP..  MAN. —  13 


194  MAGNETISM    AND   ELECTRICITY 

How  does  the  polarity  of  the  magnetism  induced  in  soft  iron 
account  for  its  attraction  by  the  magnet  ? 

e.  Repeat  the  tests  of  paragraphs  a  and  d  using  the  piece  of 
steel  instead  of  the  soft  iron.  State  the  results  and  compare 
them  with  those  obtained  with  the  iron.  Account  for  the 
differences. 

IT.  To  study  the  magnetic  action  of  magnetized  filings 
when  arranged  with  tJieir  like  poles  pointing  in  the  same 
direction,  and  when  lying  irregularly f  to  observe  the  effect 
of  twisting  a  magnetized  wire. 

a.  Shake  the  filings  away  from  the  bottom  of  the  test  tube ; 
then,  with  the  tube  in  a  horizontal  position,  jar  them  back  by 
gentle  and  repeated  tapping  of  the  bottom  of  the  tube  against 
the  north  pole  of  the  magnet.     The  filings  are  thus  brought 
into  their  new  positions  under  the  action  of  the  north  pole 
of  the  magnet.  'The  filings  at  the  bottom  of  the  test  tube 
should  now  repel  one  pole  of  the  magnetic  needle.     If  they 
do,  what  is  the  polarity  at  this  end  of  the  tube  ?     If  they  do 
not,  repeat  the  experiment  till  successful. 

Apply  the  same  test  to  the  filings  again  after  thoroughly 
shaking  the  tube.  Remember  that,  where  the  magnetic  action 
is  weak,  repulsion  is  the  only  test  for  magnetism  not  induced 
by  the  presence  of  the  testing  needle  itself. 

State  and  explain  the  result  of  the  test. 

b.  Briefly  discuss  the  experiment  as  an  illustration  of  the 
theory  of  magnetism. 

c.  Magnetize  the  long  piece  of  wire  by  rubbing  it  from  the 
center  to  one  end  with  one  pole  of  the  magnet  and  to  the  other 
end  with  the  other  pole.     Test  the  amount  of  magnetism  de- 
veloped in  the  wire  by  the  quantity  of  filings  that  it  will  pick 
up.     Twist  the  wire  both  ways  a  few  times,  and  again  test  its 
magnetism. 

How  does  the  theory  of  magnetism  account  for  the  observed 
effect  of  the  twisting  ? 


THE   MAGNETIC   FIELD  195 

EXERCISE   65.     THE   MAGNETIC   FIELD 

References.— Hoadley,  299-300 ;  Carhart  and  Cluite,  374-376. 

To  study  and  to  make  maps  of  magnetic  fields,  show- 
ing the  shape  and  direction  of  the  lines  of  force. 

Apparatus.  —  Magnetic  needle  suspended  by  a  thread ;  two 
bar  magnets ;  piece  of  thick  cardboard  about  9  by  11  in. ;  board 
of  the  same  size  or  larger,  with  parallel  grooves,  about  an  inch 
apart,  in  which  the  magnets  will  lie  flush  with  the  surface ; 
blue-print  paper ;  iron  filings  in  pepper  box  or  other  sifter. 

a.  Move  the  magnetic  needle  around  and  over    a   pole    of 
a  bar  magnet  upon  the  table.     Observe  the  behavior  of  the 
needle.     Do  the  same  at  the  other  end  of  the  magnet  and 
compare  results.     Move  the  needle  over  the  magnet  from  end 
to  end,  also  along  the  sides  of  the  magnet  and  beyond  the  ends ; 
and  observe  the  direction  of  the  needle  in  all  positions. 

b.  Place  a  magnet  in  a  groove  of  the  board.      If  blue-print 
paper  is  provided,  fasten  a  piece  of  it  by  means  of  rubber 
bands  to  the  cardboard  with  the  prepared  side  up,  and  place  it 
over  the  magnet.     Keep  unused  paper  in  the  dark.     Sprinkle 
filings  thinly  and  evenly  over  the  paper,  and  gently  tap  the 
card  with  the  finger  while  holding  it  in  place. 

Move  the  magnetic  needle  about  just  above  the  paper.  How 
does  it  set  itself  with  respect  to  the  lines  of  filings  ? 

Why  are  these  lines  called  lines  of  force  ? 

Lift  the  cardboard  vertically  from  the  magnet,  and  place  it 
in  the  sun  for  about  three  minutes.  If  the  sun  is  not  shining, 
place  it  in  the  strongest  daylight  available  for  five  minutes  or 
more.  Return  the  filings  to  the  box,  and  wash  the  paper  im- 
mediately by  moving  it  about  in  clean  water  or  letting  water 
run  over  it  from  the  faucet  for  a  few  minutes.  After  it  has 
dried,  fasten  it  in  your  notebook.  If  left  to  dry  in  the  labora- 
tory till  the  following  day,  write  your  name  on  it  for  identi- 
fication. 


196  MAGNETISM   AND   ELECTRICITY 

If  blue-print  paper  is  not  provided,  sprinkle  the  filings  on 
the  cardboard  and  make  a  drawing  of  the  magnet  and  its  field 
in  your  notebook,  sketching  in  a  number  of  the  lines  shown 
by  the  filings. 

c.  In  the  same  way  make  a  blue  print  (or  drawing)  of  the 
magnetic  field  about  two  bar  magnets  placed  parallel  in  the 
grooves  with  like  poles  turned  in  the  same  direction.     Mark 
the  poles  N  and  S  in  the  print  or  drawing. 

Repeat  with  like  poles  in  opposite  directions. 
Do  you  find  in  these  prints  lines  of  force  connecting  unlike 
poles  ?     Do  you  find  them  connecting  like  poles  ? 

d.  Place  arrowheads  on  some  of  the  lines  of  force  in  all 
of  the  prints  or  drawings,  pointing  from  the  north  pole.     What 
do  these  arrowheads  denote  ?     (See  references.) 

EXERCISE  66.     THE   SINGLE-FLUID   CELL 

References.  —  Hoadley,  346-351;  Carhart  and  Chute,  428- 
431  and  433-435  ;  Sanford,  pp.  281-282. 

Apparatus.  —  Two  test  tubes ;  small  pieces  of  zinc  and  cop- 
per ;  bottle  of  dilute  sulphuric  acid ;  matches ;  tumbler  of  dilute 
sulphuric  acid  (about  one  part  by  volume  of  concentrated  acid 
to  twenty  parts  of  water) ;  strip  of  copper  and  two  of  zinc, 
one  amalgamated,  with  a  wire  soldered  to  each  strip ;  wooden 
block  to  hold  the  strips  (Fig.  84)  ;  double  connector;  magnetic 
needle. 

I.  To  study  the  chemical  action  of  dilute  sulphuric 
acid  on  zinc  and  copper. 

a.  Put  a  few  pieces  of  zinc  in  a  test  tube  and  pour  on  them 
a  little  sulphuric  acid.  Invert  the  other  test  tube  over  the 
first  to  collect  the  gas  liberated.  Observe  the  action.  After  a 
minute  or  two,  remove  the  upper  test  tube,  keeping  it  still  in- 
verted, and  hold  a  lighted  match  to  its  mouth.  There  should 


THE   SINGLE-FLUID    CELL 


197 


be  a  small  explosion,  indicating  the  presence  of  a  combustible 
gas.  This  is  hydrogen,  which  has  been  set  free  from  the  acid 
by  the  zinc.  Return  the  acid  to  the  bottle  and  remove  the 
zinc. 

b.  Try  the  action  of  the  acid  on  some  pieces  of  copper  in  a 
test  tube.     State  the  result. 

II.  To  study  the  action  of  a  simple  voltaic  cell  with 
unarnal^amated  zinc;  and  to  test  the  presence  of  a 
current  by  its  magnetic  action  on  a  compass  needle. 

a.  Insert  the  strips  of  copper  and  unamalgainated  zinc  into 
the  slits  of  the  supporting  block  and  place  them  in  the  tumbler 
of  acid.  (The  darker  piece  of 
zinc  is  the  unamalgamated  one.) 
Avoid  all  metallic  connections 
between  the  strips.  Observe  for 
a  short  time  and  record  what 
happens  at  the  surface  of  each 
strip.  Avoid  inhaling  the  escap- 
ing gas,  as  it  contains  irritating 
impurities. 

6.  Connect  the  copper  and 
zinc  strips  outside  the  cell  by 
means  of  the  wire  and  connec- 
tor. (Connections  must  always 
be  firmly  made  to  the  bare 
wire.  The  current  will  not  pass 
through  the  insulation  covering 
the  wire.)  Observe  and  record 
what  happens  at  the  surface  of 
each  strip.  Explain  briefly  after  consulting  the  references. 

c.  Turn  and  bend  the  connecting  wire  so  that  a  portion  of  it 
is  horizontal  and  extends  north  and  south.     Hold  the  magnetic 
needle  very  close  to  this  portion  of  the  wire  and  just  below  it. 
Observe  whether  the  needle  comes  to  rest  in  a  different  direc- 


FIG.  84. 


108  MAGNETISM    AND    ELECTRICITY 

tion  when  brought  near  the  wire.  ,  If  there  is  a  deflection  of 
the  needle,  it  is  evidence  of  an  electric  current  in  the  wire. 
How  is  the  deflection  of  the  needle  affected  by  holding  it  just 
above  the  wire  ?  Remove  the  zinc  from  the  acid  at  once. 

III.  To  study  the  action  of  a  simple  voltaic  cell  with 
amalgamated  zinc. 

a.  Place  the  amalgamated  zinc  in  the  cell  instead  of  the 
unamalgamated,  and  observe  the  action  at  its  surface  with  the 
wires  disconnected.      Compare  with  the  action  on  unamalga- 
mated zinc  under  the  same  conditions,  and  account  for  the  dif- 
ference.    (See  references.) 

b.  Connect  the  strips  with  a  wire  and  observe  what  happens 
at  their  surfaces.     Compare   with  the   results   of  II  b.  and 
account  for  the  difference. 

c.  Test  the  presence  of  a  current  in  the  wire  by  the  de- 
flection of  the  needle,  as  before. 

Disconnect  the  wire  and  remove  the  strips  from  the  acid. 

EXERCISE   67.     THE   MAGNETIC   EFFECTS   OF  A 
CURRENT 

References.  — Hoadley,  355  and  371-376 ;  Carhart  and  Chute, 
433-434,  441,  and  452-458. 

Apparatus.  —  Magnetic  needle  (see  Exercise  64) ;  wire 
rectangle  of  several  turns  and  about  25  cm.  in  diameter 
(Fig.  85) ;  Grenet  cell  (or  other  cell  that  will  furnish  a  current 
of  several  amperes);  cardboard;  tangent  galvanometer  with- 
out the  needle  (or  a  circular  coil  of  15  to  20  turns  and  about 
15  cm.  diameter)  ;  helix  on  cardboard  and  soft  iron  rod  for  core 
(Fig.  87);  electromagnet  consisting  of  the  core  of  the  helix 
and  a  long  coil  of  small  wire  of  about  200  turns  wound  on  a 
cardboard  tube ;  nail  or  other  piece  of  iron ;  two  double  con- 
nectors. 


THE   MAGNETIC   EFFECTS   OF  A  CURRENT  199 

I.  To  find  the  shape  and  direction  of  the  lines  of  force 
of  a  current  in  a  straight  wire;  and  to  find  the  relation 
between  the  direction  of  the  lines  of  force  and  the  direc- 
tion of  the  current. 

a.  Support  the  wire  rectangle  in  a  vertical  plane  extending 
north  and  south,  and  connect  it  with  the  cell.  If  a  Grenet 
cell  is  used  the  circuit  is  closed  by  lowering  the  zinc  into  the 
liquid.  Keep  the  zinc  out  of  the  liquid  as  much  as  possible. 
Place  the  cardboard  in  position  around  ,„____ ,__ 
one  of  the  sides  of  the  rectangle  and 
close  the  circuit.  Sift  filings  upon  the 
cardboard,  and  tap  it  gently  until  the 
filings  are  arranged  in  distinct  lines. 

Determine  from  the  battery  connec- 
tions whether  the  current  is  flowing  up 
or  down  in  the  wires  through  the  card- 
board. (It  flows  through  the  wire  from 
the  carbon  to  the  zinc.)  Use  the  mag-  ' 
netic  needle  to  determine  the  direction 

of  the  lines  of  force  about  the  wire  (the  direction  indicated  by 
the  north  pole  of  the  needle).  Raise  the  zinc.  Draw  a  figure 
in  perspective,  showing  the  direction  of  the  current  and  the 
lines  of  force. 

6.  Grasp  the  wire  with  the  right  hand,  with  the  thumb  ex- 
tended and  pointing  in  the  direction  in  which  the  current  was 
flowing.  Do  the  fingers  point  in  the  direction  of  the  lines  of 
force  round  the  wire  or  in  the  opposite  direction  ?  Remember 
this  relation.  It  is  called  the  right-hand  rule.  State  it  in  full. 

c.  Close  the  circuit ;  and,  by  means  of  the  needle,  determine 
the  direction  of  the  lines  of  force  about  the  opposite  side  of 
the  rectangle.     Is  their  direction  in  agreement  with  the  right- 
hand  rule  ? 

d.  Observe  the  deflection  of  the  needle  when  held  just  above 
the  upper  side  of  the  rectangle  and  when  held  just  below  it. 
Are  the  deflections  in  agreement  with  the  rule  ? 


200 


MAGNETISM   AND   ELECTRICITY 


Several  turns  are  used  in  the  rectangle  merely  to  intensify 
the  effect.  With  the  same  current,  the  magnetic  field  about  a 
single  wire  would  be  weaker,  but  otherwise  the  same. 

II.  To  find  the  shape  and  direction  of  the  lines  of 
force  about  a  coil  of  wire  in  which  a  current  is  flow- 
ing ;  and  to  find  the  relation  between  tJie  direction  of  the 
cuwcnt  and  the  direction  of  the  lines  of  force  at  the 
center  of  the  coil- 

a.  Connect  the  cell  to  the  two  binding  posts  of  the  galva- 
nometer between  which  all  the  turns  of  the  coil  are  included. 
Adjust  the  cardboard  to  the  middle  of  the  coil  (Fig.  80),  and 
close  the  circuit.     Sprinkle  filings  on  the  cardboard,  and  tap 
with  the  finger.     Determine  with  the  needle  the  direction  of 

the  lines  of  force  through  the 
coil.  Eaise  the  zinc.  If  pos- 
sible, find  from  the  galvanom- 
eter and  battery  connections 
and  the  winding  of  the  coil 
the  direction  of  the  current 
through  the  coil.  If  the  wind- 
ing of  the  coil  is  not  open  to 
view,  find  the  direction  of  the 
current  from  the  right-hand 
rule  and  the  known  direction 
of  the  lines  of  force.  (The 

rule  applies  to  the  parts  of  a  curved  conductor  as  well  as.  to 
a  straight  one,  as  the  change  in  the  field  caused  by  the  bending 
of  the  conductor  does  not  affect  the  relation  expressed  by  the 
rule.)  Draw  a  figure  in  perspective  showing  the  direction  of 
the  current  and  the  lines  of  force.  Return  the  filings  to 
the  box. 

b.  As  applied   to  coils,  a  different  statement  of  the  right- 
hand  rule  is  more  convenient.     Close  the  right  hand  and  place 
it  within  the  coil   with  the  extended  thumb  pointing  in  the 


FIG. 


THE   MAGNETIC    EFFECTS   OF   A   CURRENT  201 

direction  of  the  lines  of  force  through  the  coil.  Do  the  fingers 
point  in  the  direction  of  the  current  through  the  coil  or  in  the 
opposite  direction  ?  State  the  rule  in  full. 

III.  To  find  the  shape  and  direction  of  the  lines  of 
force  within  and  about  a  helijo ;  and  the  relation  betwee?i 
the  polarity  of  the  helijo  and  the  direction  of  the  current 
round  it. 

Pass  the  current  through  the  helix  on  the  cardboard,  and 
use  filings  to  show  the  lines  of  force.  Determine  the  direction 
of  the  current  round  the  helix  from  the  battery  connections, 
and  the  polarity  of  the  helix  by  means  of  the  needle.  (As  the 


FIG.  87. 

field  of  the  coil  is  weak,  it  should  lie  east  and  west  to  dis- 
tinguish its  action  from  that  of  the  stronger  field  of  the 
earth.)  Draw  a  figure  in  perspective  showing  the  polarity  of 
the  helix  and  the  direction  of  the  current. 

Are  the  observed  relations  in  agreement  with  the  right-hand 
rule  for  coils  ? 

IV.    To  jnalce  an  electromagnet  and  test  its  strength. 

a.  Again  pass  the  current  through  the  helix,  and  place  the 
soft  iron  core  inside  it.     Study  the  field  with  filings  and  deter- 
mine the  polarity  as  before.     What  is  the  effect  of  the  core  ? 

b.  Connect  the  coil  of  the  electromagnet  to  the  battery  and 
insert  the  iron  core.     Test  the  strength  of  the  electromagnet 


202' 


MAGNETISM  AND   ELECTRICITY 


thus  formed.  How  does  it  compare  in  strength  with  the 
permanent  magnets  previously  used  ? 

Raise  the  zinc  and  return  all  filings  to  the  box. 

48.  The  Tangent  of  an  Angle.  —  In  a  right  triangle  the  ratio 
of  one  leg  to  the  other  is  called  the  tangent  of  the  angle  opposite 

to  the  first  leg.  Thus,  in  the 
right  triangle  ABC  (Fig.  88), 
BC:AC  is  the  tangent  of 
angle  A,  and  AC:  EC  is  the 
tangent  of  angle  E\  or,  as 
usually  expressed, 

tan  A  =  BC:  AC 
FIG.  88.  and       tan  B  =  AC :  BC, 

in  which  the  abbreviation  tan  is  used  for  tangent. 
Since  triangles'  AB C  and  AB'C'  are  similar, 

BC:AC=B'C':AC'', 
and  hence  tan  A  =  B'C' :  AC'. 

From  which  it  will  be  seen  that  the  tangent  of  an  angle,  like 
its  sine  (Art.  41),  is  independent  of  the  size  of  the  triangle. 

Draw  right  triangles  with  angle  A  of  different  sizes,  the 
largest  near  90°,  and  find  whether  the  angle  and  its  tangent 
increase  proportionally.  The  comparison  is  simplified  by 
making  the  side  adjacent  to  the  angle  the  same  in  all  the 
triangles. 

As  an  angle  increases  (up  to  90°),  which  increases  the  faster, 
its  sine  or  its  tangent  ? 

What  is  the  tangent  of  45°  ? 

As  an  angle  approaches  90°,  what  value  does  its  tangent 
approach  ? 

49.  The  Tangent  Galvanometer.  —  In  Exercise  67  it  was  found 
that  the  lines  of  force  of  the  current  in  a  tangent  galvanometer 
for  circular  coil)  were  approximately  straight  near  the  middle 


THE  TANGENT  GALVANOMETER 


203 


FlG  89 


of  the  coil,  where  the  needle  of  the  instrument  is  placed,  and 
were  at  right  angles  to  the  plane  of  the  coil.  In  using  the 
instrument  it  is  always  set  with  the  plane  of  the  coil  in  the 
magnetic  rneridan;  hence,  at  the  center  of  the  coil,  the  lines  of 
force  of  the  current  are  at  right  angles  to  those  of  the  earth's 
field. 

In  Fig.  89  let  0  denote  the  center  of  the  galvanometer  coil, 
ON  the  strength  and  direction  of  the  earth's  field,  and  OC 
the  strength  and  direction 

of  the  field  of  the  current     N, —     — *A      N, ^A 

in  the  coil.    The  direction 

of  the  resultant  of  these 

forces  is   OA\    which   is 

therefore  the  direction  of 

the  resultant  force  upon 

the  north  pole  of  the  needle.     Since  the  resultant  force  upon  the 

south  pole  is  equal  and  opposite  to  that  upon  the  north  pole, 

the  needle  will  come  to  rest  in  the  line  OA. 

Now  suppose  the  strength  of  the  field  of  the  current  to  be 
doubled  (which  would  be  the  case  if  the  current  were  doubled, 
since  a  current  is  measured  by  its  magnetic  effect).  ON  re- 
mains the  same,  as  the  earth's  field  is  not  changed.  The  angle 
of  deflection  is  increased,  but  is  not  doubled.  In  fact,  if  at 
first  OC  is  equal  to  ON,  angle  NO  A  is  45°,  and  can  never  be 
quite  doubled  however  great  OC  may  become.  Clearly  the 
currents  are  not  proportional  to  the  deflections  that  they 
produce. 

Let  C  and  C1  denote  two  currents  through  the  same  number 
of  turns  of  the  same  galvanometer,  and  let  OC  and  OC' 
(or  NA  and  NB,  Fig.  90)  denote  the  strengths  of  their  fields 
(at  the  center  of  the  galvanometer).  ON,  as  before,  is  the 
strength  of  the  earth's  field.  Then  angles  a  and  a'  are  the 
deflections  caused  by  C  and  C'  respectively. 

Since  the  currents  are  measured  by  their  magnetic  effects, 
C :  C1  :  :  NA  :  NB. 


204 


MAGNETISM   AND   ELECTRICITY 


C 

FIG.  90. 


If  the  terms  of  the  second  ratio  are  divided  by  the  same 
quantity  OJV,  the  value  of  the  ratio  is  not  altered  j  hence 

r.^..NA.NB 

"ON' ON1 

or  C:  C'  :  :  tan  a:  tan  a'.  (1) 

That  is,  the  currents  are  proportional  to  the  tangents  of  the 

angles  of  deflection.  This  is  why 
the  instrument  is  called  a  tangent 
galvanometer. 

To  illustrate:  A  current  (C) 
causes  a  deflection  of  50°  and 
another  current  (C")  a  deflection 
of  25°.  By  referring  to  the  Table 
of  Tangents  in  the  Appendix, 

it  is  found  that  fan  50°  =  1.19  and  tan  25°  =  .466. 

Hence  C:  C' :  :  1.19  :  .466; 

from  which  C=2.55  C'. 

While  the  tangent  galvanometer  may  be  thus  used  to  com- 
pare currents,  it  does  not  give  their  value  in  amperes  (unless 
provided  with  a  calibrated  tangent  scale).  The  deflections  can 
be  reduced  to  amperes ;  but  the  methods  by  which  this  is  done 
lie  beyond  the  scope  of  an  elementary  course. 

Is  the  deflection  affected  by  the  strength  of  the  magnetic 
needle  ? 


EXERCISE  68.  THE  TANGENT  GALVANOMETER 

References.  —  Hoadley,  356-357  and  382 ;  Carhart  and  Chute, 
460,  464-466,  and  471-472 ;  Slate,  230. 

To  learn  to  use  a  tangent  galvanometer;  and  to  find 
the  relation  between  the  number  of  turns  of  the  coil 
through  which  the  current  is  sent  and  the  strength  of 
the  magnetic  field  of  the  current. 


THE  TANGENT  GALVANOMETER          205 

Apparatus.  —  A  tangent  galvanometer  with  three  different 
connections  (usually  for  5,  10,  and  15  turns  of  the  coil) ;  con- 
stant cell  (gravity  or  Daniell) ;  wires  for  connections ;  two 
double  connectors ;  two  "  compensating  coils "  having  resist- 
ances equal  respectively  to  5  and  10  turns  of  the  galvanometer 
coil;  another  resistance  coil,  if  necessary  (see  the  experi- 
ment). 

[The  coils  are  easily  made  by  winding  the  necessary  amount  of  wire 
on  spools.  They  serve  the  purpose  better  than  a  resistance  box  at  this 
stage  of  the  work.] 

a.  Turn  the  galvanometer  so  that  the  ends  of  the  pointer 
attached  to  the  needle  (or  the  ends  of  the  needle,  if  it  does  not 
carry  a  pointer)  stand  as  nearly  as  possible  at  zero.  Tap  the 
galvanometer  gently  with  the  finger  to  overcome  friction,  which 
might  otherwise  hold  the  needle  in  a  wrong  position.  (Always 
observe  this  precaution  before '  reading  a  deflection.)  When 
adjusted  as  directed,  the  coil  of  the  galvanometer  will  be  in 
the  plane  of  the  magnetic  meridian.  Care  must  be  taken  not 
to  disturb  this  adjustment  during  the  exercise. 

The  three  binding  posts  on  the  galvanometer  make  possible 
three  different  connections  by  which  the  current  is  sent  through 
the  number  of  turns  of  the  coil  marked  on  the  base.  Connect 
the  cell  with  the  galvanometer  so  as  to  include  all  the  turns 
of  the  coil ;  and,  if  necessary,  introduce  resistance  into  the  cir- 
cuit to  prevent  a  deflection  exceeding  65°.  Do  not  use  the 
"  compensating  coils  "  for  this  purpose.  Any  resistance  intro- 
duced here  must  be  retained  in  the  circuit  throughout  the  exercise. 
After  tapping  the  galvanometer,  read  the  position  of  both  ends 
of  the  pointer.  In  taking  the  reading,  close  one  eye  and  look 
vertically  down  upon  the  pointer  with  the  other.  E/ead  to  the 
nearest  .5°.  Keverse  the  connections  with  the  galvanometer 
(using  the  commutator,  if  one  is  provided),  thus  causing  a 
deflection  in  the  opposite  direction,  and  take  readings  as  be- 
fore. Take  the  average  of  these  four  readings  as  the  true 
deflection. 


206 


MAGNETISM   AND    ELECTRICITY 


b.  Connect  with  the  galvanometer  so  as  to  send  the  current 
through  the  next  small  number  of  turns ;  and,  in  addition  to 
the  resistance  previously  included  (if  any),  include  the  smaller 
compensating  coil,  whose  resistance  is  equal  to  the  resistance 
of  the  turns  of  the  galvanometer  coil  that  have  been  dropped 
from  the  circuit. 

The  purpose  of  this  adjustment  is  to  keep  the  resistance  of 
the  circuit  the  same  as  before.  For  a  constant  current  is 
required  throughout  the  exercise,  and  this  is  furnished  by  a 
constant  cell  (E.M.F.  constant)  through  a  constant  resistance. 
(See  Ohm's  law.) 

Find  the  average  deflection  from  four  readings  as  before. 

c.  Connect  with  the  smallest  number  of  turns  of  the  galva- 
nometer, and  replace  the  smaller  compensating  coil  with  the 
larger,  leaving  the  circuit  otherwise  unchanged.     Determine 
the  deflection  as,,  before. 

d.  In  order  to  test  the  constancy  of  the  cell  throughout  the 
exercise,  connect  again  as  for  the  first  set  of  readings.     If  the 
deflection  is  appreciably  different  from  that  first  obtained,  it 
will  be  desirable  to  repeat  the  whole  exercise  if  time  permits. 
If  this  is  done,  record  both  series  of  observations,  and  com- 
pute the  results  called  for  from  both. 

Leave  the  cell  disconnected. 

FORM  OF  EECORD 


DEFLECTION 

No.  OF 

or  POINTER 

Av.  DEFLEC- 

TAN  a 

n  -f-  TAN  a 

E.  end 

W.  end 

a 



N. 

S. 

S 

N 

b 



N. 

S_ 

S. 

N. 







•\r 

S. 

N. 







THE   LAWS   OF   RESISTANCE  207 

Discussion. —  From  the  Table  of  Tangents  in  the  Appendix  find 
the  tangents  of  the  average  deflections.  The  numbers  in  the 
last  column  of  the  record  are  found  by  dividing  the  number  of 
turns  used  in  each  case  by  the  tangent  of  the  corresponding 
deflection.  These  three  quotients  should  be  equal.  If  the 
cell  is  fairly  constant,  the  per  cent  of  difference  between  the 
greatest  and  the  least  of  the  quotients  should  not  exceed  5%. 
Compute  it. 

The  equality  of  these  quotients  is  expressed  by  the  formula 

n  :  tan  a :  :  n' :  tan  a', 
or,  n  :  n' : :  tan  a  :  tan  a1. 

From  this  it  follows  that,  for  a  constant  current)  the  strength 
of  the  field  of  the  current  is  proportional  to  the  number  of 
turns  through  which  it  flows.  Compare  this  with  the  relation 

C :  C' : :  tan  a  :  tan  a', 

established  in  Art.  49.  The  latter  relation  holds  when  the  cur- 
rents are  sent  through  the  same  number  of  turns  of  the  coil.  It 
will  be  seen  that  the  same  effect  (upon  the  field  and  the  deflec- 
tion) would  be  produced  by  doubling  the  number  of  turns 
through  which  a  given  current  is  sent,  or  by  sending  twice 
the  current  through  the  original  number  of  turns ;  and  similarly 
for  an  increase  (or  decrease)  in  any  ratio. 

EXEKCISE  69.     THE  LAWS   OF  KESISTAISTCE 

References.  —  Hoadley,  356-357  and  379-380;  Carhart  and 
Chute,  460-462  and  464-466 ;  Slate,  234-235. 

Apparatus.  —  A  low-resistance  galvanometer;  constant  cell; 
board  with  wires  (Fig.  91) ;  sliding-contact  piece ;  meter  rod ; 
connecting  wires. 

[For  the  resistances,  stretch  on  a  long  board,  between  binding  posts, 
three  bare  German  silver  wires  of  different  sizes  (Nos.  25  to  30  are  suit- 
able) and  about  a  meter  long,  and  8  to  10  in.  of  insulated  copper  wire 
of  the  same  size  as  one  of  the  German  silver  wires.  Steel  wire  may  be 


208  MAGNETISM   AND    ELECTRICITY 

used  instead  of  the  German  silver ;  but  it  should  be  of  smaller  sizes  or 
longer,  as  its  resistance  is  less.  For  ttte  measurement  or  comparison  of 
resistances  by  substitution  with  a  galvanometer  of  15  turns  and  a  gravity 
or  a  Daniell  cell,  resistances  between  '2  and  6  ohms  give  the  best  results  ; 
and  beyond  a  range  of  1  to  10  ohms  the  probable  error  is  large.  For  this 
exercise  and  the  following  a  tangent  galvanometer  is  not  essential.  Any 
low-resistance  galvanometer  that  gives  a  suitable  deflection  may  be  used. 
The  sliding-contact  piece  may  be  any  device  similar  to  that  used  with  a 
slide-wire  bridge.  The  diameters  of  the  wires  arid  the  length  of  the  copper 
wire  should  be  accurately  measured  and  recorded  beside  them.] 

I.  To  observe  the  effect  of  length  on  the  resistance  of  a 
conductor. 

Adjust  the  galvanometer.  (See  directions  at  the  beginning 
of  Exercise  68.)  Connect  the  cell  with  the  15  turns  of  the 
galvanometer  and  with  one  end  of  one  of  the  German  silver 
(or  steel)  wires.  Complete  the  circuit  as  shown  in  Mg.  91, 

using  the  sliding-contact  piece 
for  the  second  connection 
with  the  German  silver  wire. 
(In  diagrams  a  cell  is  usually 
represented  by  two  parallel 
lines,  as  shown  at  C  in  the 
figure,  the  short,  heavy  line 
representing  the  zinc  and  the  lighter  and  longer  line  the  carbon 
or  copper.  A  galvanometer  of  any  form  is  represented  by  a 
circle  with  a  short  line  at  the  center,  representing  the  needle, 
as  at  6r.)  Make  contact  at  different  places  along  the  stretched 
wire,  and  observe  the  effect  upon  the  deflection  of  the  gal- 
vanometer needle.  Explain.  No  readings  or  measurements 
need  be  taken. 

It  is  not  possible  by  this  method  to  determine  the  definite 
relation  between  the  resistance  of  a  wire  and  its  length,  because 
other  resistances  are  necessarily  included  in  the  circuit,  and 
they  too  affect  the  strength  of  the  current,  In  the  remainder 
of  the  exercise  it  will  be  assumed  that  the  resistance  of  a  wire 
(of  given  size  and  material)  is  proportional  to  its  length. 


THE    LAWS   OF   RESISTANCE  209 

II.  To  find  the  relative  resistance  of  (reriYian  silver  and 
copper. 

a.  Connect  the  cell  and  galvanometer  in  circuit  with  the 
copper  wire  on  the  board.      Tap  the  galvanometer  and  read 
the  deflection  of  one  end  of  the  pointer  as  nearly  to  a  tenth 
of  a  degree  as  possible.     Kecord  the  length  and  diameter  of 
the  wire  and  the  deflection. 

b.  Substitute  for  the  copper  wire  the  German  silver  wire 
of  the  same  diameter,  connecting  as  in  Part  I.     The  deflection 
must  be  in  the  same  direction  as  with  the  copper  wire.    Adjust 
the  sliding  contact  till  the  reading  of  the  same  end  of  the 
pointer  is  exactly  the  same  as  before ;  and  measure  the  length 
of  German  silver  wire  included  in  the  circuit  (that  is,  from  the 
end  where  connection  is  made  to  the   sliding  contact). 

The  resistance  of  this  portion  of  the  wire  is  equal  to  that 
of  the  copper  wire.  Why  ? 

Why  is  it  unnecessary  to  take  the  average  of  four  readings 
for  the  deflection,  as  in  the  preceding  exercise  ? 

c.  How  many  times  greater  than  this  would  be  the  resistance 
of  a  German  silver  wire  of  the  same  diameter  and  as  long  as 
the  copper  wire  ?     This  is  the  relative  resistance  of  German 
silver  and  copper  as  determined  by  your  experiment.     Com- 
pare with  the  value  given  in  Table  XIII  of  the  Appendix. 

III.  To  find  the  relation  between  the  area  of  the  cross 
section  of  a  conductor  and  its  resistance. 

a.  Include  in  the  circuit  the  whole  of  the  German  silver 
wire  of  largest  diameter,  and  read  the  position  of  one  end  of 
the  pointer  as  accurately  as  possible.     Record  the  deflection 
and  the  length  (^)  and  diameter  (dj  of  the  wire. 

b.  Substitute  the  next  smaller  German  silver  wire  for  the 
first,  using  the  sliding  contact,  and  adjust  the  contact  till  the 
deflection  is  the  same  as  before.     Eecord  the  diameter  (c?2)  of 
the  wire  and  the  length  (72)  included  in  the  circuit. 

c.  Substitute  the  smallest  wire  and  repeat. 

PHY.   LAB.  MAN. 14 


210  MAGNETISM   AND   ELECTRICITY 

FORM  OF  RECORD  FOR  PART  III 


LENGTHS  OF 
WIRE  USED 

DIAMETERS 
OF  WIRES 

KATIO  OF  RESIST- 
ANCES FOR  EQUAL 
LENGTHS  (/x) 

CROSS  SECTIONS 
OF  WIRES 

KATIO  OF 
CROSS  SEC- 
TIONS 

a 

ll  =  Cm. 

di  =  mm. 

a\  —  smin. 

b 

12=  

d.2=  

r2  :ri  =  

a2  =  

ai  :a2=—  — 

c 

h  =  

^3  —  

,.3  :ri=  

«3  =  

di  :as  =  

Discussion.  —  a.   The  lengths  /j,  12,  and  13  of  the  three  wires 
have  equal  resistances.     Why  ?     Let  r±  denote  this  resistance. 

b.  Assuming  that  resistance  is  proportional  to  length,  find 
from  these  lengths  how  many  times  rt  would  be  the  resistance 
of  lengths  of  the  second  and  third  wires  equal  to  that  of  the 
first  (7j).     Let  r2  and  rs  respectively  denote  these  resistances. 

c.  Let  aly  a2,  and  as  denote  the  areas  of  the  cross  sections  of 
the  wires.     Compute  them,     (a  =  Trd2.) 

d.  Compute  the  ratios  a-i :  a2  and  aT :  as. 

e.  Test  the  proportionality  of  the  ratios  r2 :  n  and  % :  a2 ; 
also  the  ratios  r3 :  i\  and  % :  as.     Their  difference  is  due  to 
experimental  errors,  and  should  riot  exceed  5%. 

/.    State  the  relation  in  words,  being  careful  to  iiiclude  in 
the  statement  all  the  necessary  conditions. 


EXEECISE  70.     MEASUREMENT   OF  EESISTANCE 

References.  —  The  same  as  for  Exercise  69. 

To  ineaswre  the  resistance  of  a  conductor  by  the  method 
of  substitution. 

Apparatus.  —  A  low-resistance  galvanometer ;  constant  cell ; 
connecting  wires ;  resistance  box ;  two  coils  of  unknown  re- 
sistance (from  3  to  6  ohms). 

a.  Adjust  the  galvanometer.  Complete  the  circuit  through 
one  of  the  coils  whose  resistance  is  to  be  measured  and  the 


MEASUREMENT   OF   RESISTANCE  211 

fifteen  turns  of  the  galvanometer  coil.  Tap  the  galvanometer 
gently,  and  read  one  end  of  the  pointer  as  nearly  to  one  tenth 
of  a  degree  as  possible.  A  single  reading  for  each  adjustment 
will  be  sufficient  if,  throughout  the  exercise,  the  same  end  of 
the  pointer  is  read  and  the  connections  are  made  so  as  to  have 
all  the  deflections  in  the  same  direction. 

b.  Remove  the  coil  from  the  circuit,  insert  the  resistance 
box  in  its  place,  and  remove  plugs  till  the  deflection  is  the 
same  as  for  the  unknown  resistance.     While  trying  to  get  an 
equal  deflection  with  the  resistance  box,  observe  what  is  the 
least  resistance  that  will  cause  a  visible  change  in  the  deflec- 
tion ;  and,  by  comparing  this  with  the  resistance  to  be  measured, 
estimate  the  probable  accuracy  of  your  result. 

The  total  resistance  in  ohms  introduced  by  the  coils  of  the 
box  is  the  sum  of  the  numbers  at  the  places  where  the  plugs 
have  been  removed.  This  is  equal  to  the  unknown  resistance. 
Why  ? 

c.  Measure  in  the  same  way  the  resistance  of  the  other  coil. 

d.  Measure  in  the  same  way  the  resistance  of  the  two  coils, 
connected  so  that  the  whole  current  passes  through  the  two 
coils  in  succession.     (This  is  called  connecting  in  series.)    The 
resistance  of  the  two  coils  in  series  is  the  sum  of  their  separate 
resistances.     This  will  serve  as  a  test  of  the  accuracy  of  your 
results.     Even  with  careful  experimenting,  the  error  may  be 
as  great  as  6%.     If  it  exceeds  this  limit  and  time  permits, 
repeat  the  experiment. 

This  method  of  measuring  resistance  is  instructive;  but 
other  methods  are  used  where  accuracy  is  required. 

50.    Let  C  denote  the  current  that  an  electromotive  force  E 
sends  through  a  (total)  resistance  R,  and  C'  the  current  that 
the  same  electromotive  force  sends  through  a  resistance  IV  \ 
then,  by  Ohm's  law, 


212  MAGNETISM   AND    ELECTRICITY 

Dividing  the  members  of  the  first  equation  by  the  corre- 
sponding members  of  the  second  gives 

(L^HL^E-^w 

C'~  R  ''  R'~  R' 
or  C:C'::R':R.  (1) 

This  means  that  the  current  due  to  a  given  E.M.F.  is 
inversely  proportional  to  the  resistance  of  the  entire  circuit 
(including  the  resistance  of  the  battery). 

It  is  sometimes  necessary,  as  in  the  following  exercise,  to 
consider  separately  the  different  parts  of  the  total  resistance 
of  the  circuit.  Let  r  denote  the  resistance  of  the  battery 
(internal  resistance),  g  the  resistance  of  the  galvanometer, 
and  R  the  remainder  of  the  external  resistance.  With  the 
resistance  thus  denoted  in  parts,  Ohm's  law  takes  the  form 

C-          E 

(R  +  r  +  g)' 

and  (1)  becomes 


(2) 

If  a  and  a'  are  the  deflections  caused  by  the  currents  C  and 
C'  respectively,  then 

C  :  C'  :  :  tan  a  :  tan  a'.     (Art.  49.)  (3) 

Hence,  from  (2)  and  (3), 

(R1  -h  r  4-  g)  :  (R  +  r  +  g)  :  :  tan  a  :  tan  a'.  (4) 

Note  the  fact  that  this  is  an  inverse  proportionality,  and 
that  it  is  true  only  for  a  constant  E.M.F. 

EXERCISE   71.     THE   RESISTANCE   OF   A   CELL 

To  find  the  resistance  of  a  constant  cell  (gravity  or 
Daniell}  by  the  method  of  reduced  deflection. 

Apparatus.  —  A   tangent   galvanometer    of    low   resistance  ; 
gravity  or  Daniell  cell  ;   resistance  box. 


THE   RESISTANCE   OF   A   CELL 


213 


a.  Adjust  the  galvanometer,  and  connect  it  in  circuit  with 
the  cell  and  the  resistance  box.     Connect  with  the  number  of 
turns  of  the  galvanometer  that  gives  a  deflection  nearest  to 
50°  to  60°  when  no  resistance  is  introduced  in  the  box,  and  use 
only  this  connection  throughout  the  exercise. 

Let  g  denote  the  resistance  of  the  number  of  turns  of  the 
galvanometer  used,  and  record  its  value  as  given  you.  Let  r 
denote  the  (unknown)  resistance  of  the  cell ;  R,  R',  and  R" 
the  resistances  successively  introduced  in  the  resistance  box ; 
and  a,  a',  and  a"  the  corresponding  average  deflections  of  the' 
galvanometer.  The  resistance  of  the  connecting  wires  may  be 
neglected. 

With  no  resistance  introduced  in  the  box  (R  =  0).,  read  the 
position  of  both  ends  of  the  pointer  as  accurately  as  possible; 
reverse  the  current  (using  the  commutator,  if  one  is  provided) 
and  read  again. 

Repeat  with  R'  =  2  ohms  and  again  with  R"  =  4  ohms. 

b.  The  resistance  of  the  cell  can  be  computed  by  substi- 
tuting in  (4)  of  Art.  50  any  two  pairs  of  values  of  tana  and  R 
and  the  value  of  g,  and  solving  for  r. 

Compute  r  from  a,  a',  R,  and  R'. 

c.  Compute  r  from  a,  a",  R,  and  R". 

d.  Compute  r  from  a',  a",  R',  and  R1'. 

e.  These  three  independent  determinations  of  r  should  agree 
within  5  %.     Find  the  greatest  per  cent  of  difference  between 
them. 

FORM  OF  RECORD 


Box  RE- 

DEFLECTION 

OF    POINTE 

a 

AVERAGE 

SISTANCE 

E.  end 

W.  end 

E.  end 

W.  end 

DEFLECTION 

R    —  0 

N 

05 

<^ 

N' 

E'  —  2 

a 

R"  —  4 

ntl 

21  I  MAGNETISM    AND    ELECTRICITY 

51.  Let  C  denote  the  current  that  an  electromotive  force  E 
sends  through  a  (total)  resistance  R,  and  C'  the  current  that 
an  electromotive  force  E'  sends  through  the  same  resistance; 

then  (7  =  |  and  C'=^- 

H  K 


From  which 


C:C'::E:E'  (1) 


That  is,  /or  a  constant  resistance,  the  current  is  proportional 
to  the  E.  M.  F. 

From  this  and  equation  (1)  of  Art.  49  it  follows  that 

E  :  E'  :  :  tan  a  :  tan  a1.  (2) 

Thus,  if  two  batteries  are  separately  connected  with  a  tan- 
gent galvanometer  and  the  total  resistance  of  the  two  circuits 
is  the  same,  the  E.M.F.  of  the  two  batteries  will  be  proportional 
to  the  tangents  of  the  angles  of  deflection. 


EXEECISE   72,.     THE    ELECTROMOTIVE    FOKCE    OF 

CELLS 

References.  —  Hoadley,  381  and  394 ;  Carhart  and  Chute,  465. 

To  find  the  electromotive  force  of  cells  by  the  method 
of  constant  resistance. 

Apparatus.  —  A  tangent  galvanometer  of  high  resistance ; 
gravity  or  Daniell  cell ;  one  or  more  cells  for  the  measurement 
of  their  E.M.F. 

[The  galvanometer  should  have  a  resistance  of  at  least  100  ohms  ;  the 
greater  the  better.  The  E.M.F.  of  the  gravity  or  Daniell  cell  should  be 
determined  with  a  voltmeter  by  the  teacher  from  day  to  day,  and  marked 
on  the  cell.  ] 


THE  ELECTROMOTIVE  FORCE  OF  CELLS 


215 


a.  Adjust  the  galvanometer,  and  pass  the  current  from  the 
gravity  or   Daniell   cell  through  it.     Read  both  ends  of  the 
pointer,  reverse  the  current  and  again  read.    Record  the  E.M.F. 
of  the  cell. 

Find  in  the  same  way  the  deflection  caused  by  each  of  the 
other  cells  provided. 

b.  The  resistance  of  the  galvanometer  is  very  large  compared 
with  the  other  resistances  of  the  circuit ;  hence  the  difference 
in  the  resistances  of  the  cells  may  be  neglected,  and  the  total 
resistance  of  the  circuit  may  be  regarded  as  constant  through- 
out the  exercise. 

Hence  the  E.M.F.  of  each  of  the  cells  can  be  computed  by 
substituting  in  (2),  Art.  51,  the  tangent  of  the  deflection  caused 
by  it  (tan  a')  and  by  the  cell  of  given  E.M.F.  (tan  a)  and  the 
E.M.F.  (E)  of  the  latter. 

FORM  OF  RECORD 


KIND  OF  CELL 

DEFLECTION 

Av.  DEFLEC- 
TION (a) 

TAN  a 

E.M.F. 

E.  end 

W.  end 

Gravity 

N. 

S. 

S. 

N 

f  re\     c\v\\ 

\  given; 

Leclanche" 

N. 

S. 

S. 

N. 







52.  Let  E  be  the  E.M.F.  that  sends  a  current  C  through 
a  resistance  R,  and  E1  the  E.M.F.  that  sends  an  equal  current 
through  a  resistance  R'  ;  then 

r_E_E' 
-- 


Hence  E:E'::R:R'  (1) 

This  means  that,  to  maintain  a  given  current,  the  E.M.F. 
must  be  proportional  to  the  resistance. 


21(3  MAGNETISM   AND   ELECTRICITY 

EXERCISE   722.      THE   ELECTROMOTIVE   FORCE   OF 

CELLS 

References.  —  Hoadley,  381  and  394  ;  Carhart  and  Chute,  465. 

To  find  tlw  electromotive  force  of  cells  by  tlie  method 
of  equal  deflections. 

Apparatus.  —  A  high-resistance  galvanometer  (not  neces- 
sarily a  tangent  instrument)  with  its  resistance  marked  on 
it,  or  an  astatic  galvanometer  of  low  resistance  ,  resistance 
box;  gravity  or  Daniell  cell  with  its  E.M.F.  marked  on  it; 
one  or  more  cells  for  the  measurement  of  their  E.M.F. 

a.  Adjust  the  galvanometer  and  connect  with  the  gravity  or 
Daniell  cell,  including  the  resistance  box  in  the  circuit  ;  but 
before  closing  the  circuit  introduce  a  high  resistance  in  the 
box  to  avoid  the  danger  of  passing  too  large  a  current  through 
the  galvanometer.     Adjust  the  resistance  in  the  box  so  that 
the  deflection  is  between  40°  and  50°,  and  read  one  end  of  the 
pointer  as  accurately  as  possible. 

Let  R  denote  the  resistance  introduced  in  the  box,  and  g  the 
resistance  of  the  galvanometer.  If  R  -f-  g  is  more  than  100 
ohms,  the  battery  resistance  may  be  neglected. 

b.  Substitute  another  cell  for  the  one  just  used,  and  adjust 
the  resistance  in  the  box  so  that  the  deflection  of  the  same 
end  of  the  pointer  in  the  same  direction  is  equal  to  the  first 
deflection. 

Let  R'  denote  the  resistance  introduced  in  the  box,  and  E' 
the  E.M.F.  of  the  cell  ;  then,  since  the  currents  were  equal,  we 
have  from  (1),  Art.  52, 


From  which  compute  the  E.M.F.  of  the  cell. 

c.    Find  in  the  same  way  the  E.M.F.   of  the   other   cells 


provided. 


ARRANGEMENT   OF   CELLS 


217 


FORM  OF  RECORD 

Deflection  for  each  adjustment  =  - 
Resistance  of  galvanometer  (g)  =  —    —  ohms 


KIND  OF  CELL 

Box  RESIST- 
ANCE (H) 

R+g 

E.M.F. 

Gravity 
Leclanch£,  etc. 







EXERCISE   73.     ARRANGEMENT   OF   CELLS 

References.  —  Hoadley,  358-362;  Carhart  and  Chute,  467- 
470 ;  Sanf ord,  pp.  318-319. 

To  find  when  cells  should  be  connected  in  parallel  and 
when  in  series  to  obtain  the  larger  current. 

Apparatus.  —  A  tangent  galvanometer  of  low  resistance ;  two 
gravity  or  Daniell  cells  (as  nearly  alike  as  possible  in  E.M.F. 
and  resistance)  ;  resistance  box ;  two  double  connectors ;  con- 
necting wires. 

a.  Connect  one  of  the  cells  with  the  resistance  box  and 
ten  turns  of  the  galvanometer  coil,  after  adjusting  the  galva- 
nometer.    Throughout  the  exercise  make  connections  for  de- 
flection in  the  same  direction,  and  read  the  same  end  of  the 
pointer.     If    care    is    taken    not    to   disturb   the   position   of 
the  galvanometer,  this  single  reading  of  each  deflection  (to 
the  nearest  degree)  will  be  sufficient. 

Read  the  deflections  for  resistances  of  0,  1,  2,  4,  10,  and 
20  ohms  in  the  resistance  box. 

b.  Substitute  the  other  cell  and  repeat.      If,  however,  the 
deflections  with  the  second  cell  are  within  one  or  two  degrees 
of  the  first,  this  part  may  be  omitted. 


218 


MAGNETISM   AND    ELECTRICITY 


c.  Repeat  with  the  same  series*  of  resistances  and  the  two 
cells  in  parallel.     Make  a  diagram  of  the  connections. 

d.  Repeat  with  the  cells  in  series.     Make  a  diagram  of  the 
connections. 

FORM  OF  RECORD 


DKKLECTION  WITH 

DEFLECTION  FOR  CELLS 

AV.  FOR  THE 

Two  CELLS 

ONE  CULL 

OTHER  CELL 

Ix  PARALLEL 

IN  SKUIF.S 

Q 

1 

2 











etc. 

TANGENTS  FOR  CELLS 

TAXGEXT  OF  AVERAGE  OF 

SINGLE  CELL 

IN  PARALLEL 

IN  SERIES 

0 

1 







2 







etc. 

Discussion.  —  a.  If  the  deflections  were  found  for  both  cells 
separately,  find  their  average  deflection  for  each  resistance. 
Find  the  tangents  of  these  average  deflections  and  of  the 
deflections  with  the  cells  in  parallel  and  in  series,  and  record 
as  indicated. 

b.  Since  the  currents  are  proportional  to  the  tangents  of  the 
deflections,  the  relative  advantage  of  joining  the  cells  in  series 
and  in  parallel  for  the  different  resistances  is  shown  at  once  by 
the  tangents. 

Within  what  limits  does  connection  in  parallel  give  the 
larger  current? 


INDUCED    CURRENTS  219 

c.  Does  the  advantage  of  connection  in  series  become  more 
or  less  marked  as  the  external  resistance  is  increased  ? 

d.  Let  r  denote  the  resistance  and  E  the  E.M.F.  of  each  of 
the  cells,  R  the  external  resistance,  and  C  the  current.     Using 
these  letters,  write  the  formula  for  the  current,  —  (1)  when  the 
two  cells  are  connected  in  parallel ;  (2)  when  they  are  con- 
nected in  series. 

With  the  aid  of  these  formulas  show :  — 

e.  Why  two  cells  in  series  are  but  little  better  than  one 
when  the  external  resistance  is  a  small  fraction  of  an  ohm. 

/.  Why  two  cells  in  parallel  are  but  little  better  than  one 
when  the  external  resistance  is  20  ohms. 

g.  Why  two  cells  in  series  give  nearly  twice  the  current 
through  20  ohms  that  a  single  cell  does. 

EXEKCISE   74.     INDUCED   CURRENTS 

References.  —  Hoadley,  397 ;  Carhart  and  Chute,  480-482 ; 
Slate,  232 ;  Sanford,  p.  291. 

Apparatus.  —  An  astatic  or  D'Arsonval  galvanometer ;  in- 
duction coil  with  movable  primary  and  iron  core ;  bar  magnet ; 
connecting  wires. 

[With  a  high-resistance  induction  coil  a  D' Arson val  or  an  astatic 
galvanometer  of  high  resistance  gives  the  best  results.  For  a  low-resist- 
ance coil  a  low-resistance  astatic  galvanometer  is  best.] 

I.  To  find  the  direction  of  the  current  induced  in  a 
coil  of  wire  by  inserting  into  it  or  withdrawing  either 
pole  of  a  bar  magnet. 

a.  Only  very  small  currents  should  be  sent  through  an 
astatic  or  a  D'Arsonval  galvanometer;  otherwise  the  action 
on  the  needle  is  violent,  and  the  instrument  may  be  damaged. 
Since  the  galvanometer  is  to  be  used  to  determine  the  direc- 
tion of  the  induced  currents  as  well  as  to  detect  their  presence, 
it  is  necessary  first  to  determine  what  the  direction  of  the 


220  MAGNETISM   AND    ELECTRICITY 

deflection  will  be  for  a  current  of  known  direction.  With 
an  astatic  galvanometer  this  can  be  done  by  applying  the 
right-hand  rule,  if  the  connections  and  the  direction  of  winding 
are  open  to  view.  Otherwise,  proceed  as  follows :  Send  a 
current  from  one  cell  through  a  meter  or  more  of  wire,  and 

connect  the  galvanometer  as  a  shunt 
between  two  points  a  few  centi- 
meters apart  on  this  circuit  (A  and 
By  Fig.  92).  If  necessary,  increase 
the  distance  AB  till  there  is  a 
visible  deflection.  Note  its  direc- 
tion, and  find  from  the  connections  which  is  the  positive  post 
of  the  galvanometer.  If  the  current  entered  by  the  other  post, 
the  current  would  of  course  be  reversed.  Hence,  in  the  fur- 
ther work,  the  direction  of  the  deflection  will  indicate  which 
is  the  positive  post  of  the  galvanometer;  and  from  this  the 
direction  of  the  current  can  be  traced  through  the  entire 
circuit. 

&.  Adjust  the  galvanometer  and  connect  it  with  the  larger 
coil  of  wire  (called  the  secondary  coil),  placed  at  a  distance  of 
a  meter  or  more.  The  circuit  consists  only  of  the  coil,  the 
galvanometer,  and  the  connecting  wires. 

Thrust  the  north  pole  of  the  magnet  suddenly  into  the  coil, 
while  watching  the  galvanometer  needle.  Note  the  direction 
of  the  deflection.  Observe  the  effect  of  removing  the  magnet. 
Repeat  till  you  are  sure  of  the  results.  (The  galvanometer 
must  be  far  enough  away  not  to  be  directly  affected  by  the 
motion  of  the  magnet.  Test  this  by  inserting  and  withdraw- 
ing the  magnet  with  the  circuit  broken.) 

From  the  direction  of  the  deflections,  determine  the  direc- 
tion of  the  current  round  the  coil  (clockwise  or  counter  clock- 
wise, as  you  look  down  upon  it),  —  (1)  when  the  north  pole  of 
the  magnet  is  inserted ;  (2)  when  it  is  removed. 

Is  there  a  current  when  the  magnet  is  at  rest  within  the 
coil? 


INDUCED   CURRENTS  221 

c.  Applying  the  right-hand  rule,  find  the  polarity  of  the 
coil  due  to  the  current,  —  (1)  when  the  north  pole  is  inserted ; 
(2)  when  it  is  removed. 

d.  Does  the  magnetic  field  due  to  the  current  aid  or  oppose 
the  insertion  and  removal  of  the  magnet  ? 

e.  The  induced  current  that  would  magnetize  the  magnet 
with  its  existing  polarity  is  called  direct,  and  the  opposite 
current  indirect  or  inverse.    Is  the  current  direct  or  inverse,  — 
(1)  on  inserting  the  north  pole  ?  (2)  on  removing  it  ? 

/.  Eepeat  the  experiment,  inserting  and  removing  the  south 
pole ;  and  answer  the  preceding  questions  for  this  case. 

II.  To  find  the  direction  of  the  current  induced  in  a 
coil  of  wire  by  inserting  or  withdrawing  the  north  pole 
of  another  coll  in  which  a  current  is  flowing ;  and  to  find 
the  direction  of  the  induced  current  when  the  current  in 
the  inner  coil  is  started  and  when  it  is  stopped. 

a.  If  more  than  one  cell  is  provided,  connect  them  in  paral- 
lel, and  send  the  current  through  the  smaller  coil  (called  the 
primary)  in  the  direction  that  makes  the  lower  end  of  it  a 
north  pole.  The  galvanometer  is  to  be  left  connected  with 
the  secondary  coil  as  before. 

Determine,  from  the  deflection  of  the  needle,  the  direction 
of  the  current  in  the  secondary  coil,  —  (1)  when  the  north  pole  of 
the  primary  is  inserted  into  it;  (2)  when  it  is  removed. 

6.  A  current  in  the  secondary  coil  in  the  same  direction  as 
that  in  the  primary  is  called  direct,  and  a  current  in  the 
opposite  direction  is  called  inverse.  Is  the  induced  current 
direct  or  inverse,  —  (1)  when  the  coil  is  inserted  ?  (2)  when  it 
is  removed  ? 

c.  With  a  primary  coil  at  rest  in  the  secondary,  study  the 
currents  induced,  —  (1)  when  the  circuit  is  closed  through  the 
primary ;  (2)  when  it  is  broken.  (Make  and  break  the  circuit 
by  touching  the  connecting  wire  to  one  of  the  binding  posts 
and  removing  it.) 


2*22  MAGNETISM    AND    ELECTRICITY 

Compare  the  directions  of  the,  induced  currents  with  the 
directions  of  the  currents  induced  when  the  primary  coil  was 
inserted  and  removed. 

d.  Repeat  the  work  of  paragraph  c  with  the  iron  core  within 
the  primary  coil.  State  the  results  and  account  for  the  effect 
of  the  core. 

Discussion.  —  a.  State  the  different  ways  in  which  you  ob- 
tained,—  (1)  an  inverse  induced  current;  (2)  a  direct  induced 
current. 

b.  In  whatever  way  produced,  the  inverse  induced  current 
is  due  to  an  increase  in  the  number  of  lines  of  force  within 
the  coil,  and  the  direct  induced  current  to  a  decrease  in  them. 
When  the  number  of  lines  of  force  within  the  coil  is  increas- 
ing, is  the  direction  of  the  induced  current  clockwise  or  counter 
clockwise  to  an  observer  looking  at  it  in  the  direction  of  the 
lines  of  force  (that  is,  in  the  direction  in  which  the  north  pole 
of  the  magnet  or  primary  coil  points)  ? 

What  is  the  direction  of  the  induced  current  to  such  an 
observer  when  the  number  of  lines  of  force  within  the  coil  is 
decreasing  ? 

These  relations  find  application  in  the  study  of  the  dynamo 
and  motor,  and  should  be  remembered. 

c.  What  is  the  source  of  the  energy  of  the  currents  induced 
by  the  magnet  ?     (See  the  question  of  I  d.  for  a  suggestion.) 

EXERCISE   75.     THE   MOTOR 

References.  —  Hoadley,  406-415  and  439 ;  Carhart  and  Chute, 
494-500. 

Apparatus.  —  Small  motor,  preferably  one  that  can  readily 
be  taken  apart  and  put  together  again  (Fig.  93) ;  one  or  more 
cells  as  needed ;  magnetic  needle. 

I.  To  study  the  construction  and  action  of  a  small 
inotor. 


THE   MOTOR 


223 


FIG.  93. 


a.  Starting  at  either  of  the  binding  posts  of  the  motor,  trace 
the  circuit  through  the  coil  of  the  field  magnet  and  through 
the  armature  to  the  other 

post.  If  the  coils  of  the 
field  magnet  and  armature 
are  connected  in  series,  the 
motor  is  said  to  be  series 
wound;  if  they  are  con- 
nected in  parallel,  it  is 
shunt  wound.  Is  this  motor 
shunt  or  series  wound  ? 

b.  Connect  the  battery 
with  the  motor,  and  note 
by  which  binding  post  the 
current  enters  it.     While 
the  motor  is  running,  note 

the  direction  of  rotation,  and  determine  the  polarity  of  the  field 
magnet  with  the  magnetic  needle.  Is  the  relation  between  its 
polarity  and  the  direction  of  the  current  in  its  coil  in  agree- 
ment with  the  right-hand 
rule? 

c.  Disconnect  the  bat- 
tery. If  Grenet  cells  are 
used,  raise  the  zincs. 
Study  carefully  the  action 
of  the  commutator ;  and 
determine  the  direction 


FIELD  MAGNET  WIRE 


ARMATURE  WIRE 


PULLEY      TUBES  FOR 


FIG.  94. 

which  the  current  is  reversed, 
covered. 


the  coils  of  the  armature 
and  the  polarity  of  its 
core  during  one  complete 
rotation,  noting  particu- 
larly the  positions  in 
State  briefly  the  facts  dis- 


224  MAGNETISM   AND   ELECTRICITY 

d.  From  the  polarity  of  the  poles  of  the  armature  (that  is, 
the  cores  of  the  armature  coils)  in  different  positions  about  the 
axis,  account  for  the  rotation  of  the  armature. 

e.  Draw  a  simplified  diagram  of  the  motor  showing  the 
direction  of  the  current  in  the  coils  of  the  field  magnet  and 
armature,  the  polarity  of  the  field  magnet  and  of  the  poles  of 
the  armature,  and  the  direction  of  rotation. 

/.  Connect  the  battery  with  the  motor  so  that  the  current  in 
it  is  reversed  with  respect  to  its  first  direction.  Is  the  direc- 
tion of  rotation  reversed  ?  Explain. 

II.  To  take  a  small  inotor  apart  and  put  it  together 
again. 

If  the  motor  is  dissectable,  let  the  student  take  it  apart  and 
put  it  together  again ;  or,  beginning  with  it  taken  apart,  let 
him  put  it  together,  following  directions  adapted  to  the  motor 
provided  or  following  as  a  model  a  similar  motor,  finished 
(one  of  which  will  serve  for  several  students). 

EXERCISE   76.     THE   ELECTEIC   BELL  AKD 
THE   TELEGRAPH 

References.  —  Hoadley,  418-425 ;  Carhart  and  Chute,  507-515. 

I.  To  study  the  construction  and  action  of  an  electric 
bell. 

Apparatus.  —  An  electric  bell  with  rubber  tubing  on  the  clap- 
per to  deaden  the  sound ;  push  button ;  a  Leclanche  battery. 

a.  Trace  the  circuit  from  the  battery  through  the  several 
parts  of  the  bell  and  the  push  button  to  the  battery  again. 
The  metal  frame  of  the  bell  commonly  forms  a  part  of  the 
circuit.     Does  it  in  this  bell  ?     Look  carefully  for  insulation 
that  compels  the  current  to  cross  where  the  spring  that  carries 
the  armature  touches  the  end  of  a  screw. 

b.  Unscrew  the  cap  of  the  push  button  and  study  its  con- 
struction.    Describe  it. 


THE  ELECTRIC  BELL  AND  THE  TELEGRAPH    225 

c.  Connect  the  battery  to  the  bell  and  ring  it.  Observe  the 
sparks  at  the  point  where  the  spring  touches  the  screw.  What 
causes  them  ? 

Why  would  the  bell  not  ring  if  the  current  did  not  cross  at 
this  point  ? 

Briefly  explain  the  action  of  the  bell  in  connection  with  a 
simplified  drawing  showing  the  actual  connections  as  you  find 
them. 

II.  To  set  up  a  telegraph  line  and  study  the  construc- 
tion and  action  of  the  instruments. 

Apparatus.  —  A  complete  telegraph  line. 

a.  Connect  up  the  local  circuits  and  the  line  circuit,  with 
the  aid  of  the  figure  in  the  text  or  a  reference  book.     If  the 
connections  are  already  made,  trace  them  out.    Find  the  insula- 
tion in  the  key  which  keeps  the  circuit  open  except  when  the 
lever  is  depressed  or  the  switch  closed.     Find  the  insulation 
in  the  relay  which  keeps  the  local  circuit  open  when  the  line 
circuit  is  open. 

b.  Observe  that  the  line  battery  is  in  two  sections,  one  at 
each  end  of  the  line.     Which  poles  of  the  two  sections  must 
be  connected  together  ?     Why  ? 

Make  a  simplified  diagram  of  the  local  and  line  circuits  just 
as  you  find  them,  including  the  instruments  and  batteries. 

c.  Open  the  switch  at  one  end  of  the  line  and  operate  it,  at 
the  same  time  observing  the  action  of  the  sounder  and  the 
relay ;  another  student  also  observing  the  action  of  the  instru- 
ments at  the  other  station.     Now  let  the  other  student  operate 
the  key  at  his  station,  both  observing  as  before. 

d.  Try  to  operate  the  line  with  both  switches  open.     Ac- 
count for  your  success  or  want  of  success.     If  gravity  cells 
are  used,  leave  the  switches  closed  when  you  have  finished ;  if 
other  cells  are  used,  leave  the  switches  open. 

e.  Write  a  brief  statement  of  the  action  and  use  of  the  key, 
sounder,  and  relay,  and  the  use  of  the  local  and  line  batteries. 

COLEMAN'S  PHY.  LAB.  MAN.  — 15 


22l>  MAGNETISM   AND   ELECTRICITY 

EXERCISE   77.     THE   TELEPHONE 

References.  —  Hoadley,  429-431;  Carhart  and  Chute,  520- 
522, 

I.  To  study  the  construction  and  action  of  a  telephone 
receiver,  and  the  action  of  a  telephone  line  consisting  of 
two  receivers. 

Apparatus.  —  A  sensitive,  high-resistance  galvanometer 
(astatic  or  D'Arsonval) ;  two  telephone  receivers  connected 
with  long  wires ;  tuning  fork ;  rubber  mallet. 

a.  Unscrew  and  remove  the  cap  that  covers  the  disk  of  the 
receiver.     Remove  the  disk.     Describe  the  parts  exposed  to 
view.     Is  the  disk  attracted  by  the  magnet?     Of  what  ma- 
terial is  it  ? 

b.  Connect  the  receiver  with  the  galvanometer.     Place  the 
disk  in  position  on  the  receiver  and  press  it  lightly  at  the 
center  with  the  finger  so  as  to  bring  it  nearer  to  the  magnet. 
If  the  galvanometer  indicates  a  current,  account  for  it. 

c.  Connect  the  two  receivers  with  long  wires.     This  is  the 
original  form  of  the  Bell  telephone,  and  is  the  simplest  tele- 
phone  line.     Let  one    student   lightly  touch   the   stem  of  a 
sounding  tuning  fork  to  the  disk  of  one  receiver  while  another 
listens  at  the  other  receiver.     Try  speaking  to  one  another, 
using  the  receivers  alternately  as  receiver  and  transmitter. 

Explain  the  action  of  such  a  telephone  line. 

II.  To  study  the  construction  and  action  of  a  micro- 
phone. 

Apparatus.  —  A  telephone  receiver;  galvanic  cell;  two  bat- 
tery or  electric  light  carbons ;  microphone ;  tuning  fork ; 
rubber  mallet. 

a.  Connect  the  pieces  of  carbon,  the  receiver,  and  the  bat- 
tery, as  shown  in  Fig.  95,  so  that  the  circuit  is  completed  by 


THE   TELEPHONE 


227 


touching  the  carbons  together.  Place  the  receiver  to  the  ear, 
and  rub  one  carbon  lightly  upon  the  other.  The  receiver 
should  give  out  a  loud, 
rattling  sound.  The  re- 
sistance at  the  points  of 
contact  of  the  carbons 
varies  with  the  pressure. 
How  does  this  account 
for  the  sounds  from  the 
receiver  ?  Draw  a  dia- 
gram of  the  apparatus. 

In  this  experiment  the 
pieces  of  carbon  play  the  part  of  a  microphone  or  the  trans- 
mitter of  a  telephone. 

b.    The  microphone  shown  in  Fig.  96  is  a  simplified  trans- 
mitter together  with  an  induction  coil  and  receiver.     Complete 


FIG.  95. 


FIG 


the  battery  circuit  through  the  primary  coil  and  the  trans- 
mitter.    Connect  the  receiver  with  the  secondary  coil. 


228  MAGNETISM   AND    ELECTRICITY 

A  simpler  form  of  microphone  has  no  induction  coil.  If  one 
of  this  type  is  provided  instead  of  one  like  the  figure,  connect 
the  receiver  in  the  battery  circuit  with  the  microphone. 

Hold  the  receiver  to  the  ear,  and  tap  lightly  upon  the  base 
of  the  microphone  or  rub  the  finger  over  it.  Touch  a  vibrating 
tuning  fork  to  it.  Listen  to  a  watch  lying  on  the  microphone. 

Draw  a  figure  of  the  microphone  and  explain  its  action. 

III.  To  study  tlie  construction  and  action  of  a  com- 
plete telephone  line. 

Apparatus.  —  A  telephone  line  consisting  of  two  telephones 
made  for  laboratory  use. 

The  principles  of  the  telephone  have  been  covered  by  the 
preceding  experiments.  The  rest  is  a  matter  of  detail  of  con- 
struction, in  respect  to  which  telephones  differ  greatly  from 
one  another.  Study  in  detail  the  laboratory  telephone  line, 
including  the  points  covered  by  the  following  directions :  — 

a.  Trace  out  the  connections  by  which  the  bell  is  included 
in  the  line  circuit  when  the  receiver  is  on  the  hook. 

Trace  the  circuit  when  the  button  is  pushed  to  ring  the  bell 
of  the  other  telephone.  Is  the  bell  of  either  telephone  rung  by 
the  battery  of  the  same  or  the  other  telephone  ? 

b.  With  the  receiver  off  the  hook,  trace,  —  (1)  the  local  circuit 
through  the  transmitter  and  the  primary  coil ;  (2)  the  line 
circuit  through  the  secondary  coil  and  the  receiver. 

c.  What  connections  are  broken  and  what  made  by  the  lever 
when  the  receiver  is  removed  from  the  hook  ? 

d.  Study  and  use  the  line  till  you  understand  its  operation. 
Write  a  brief  description  of  this  telephone  line,  and  explain 
its  operation,  including  any  points  not  covered  by  the  directions. 


APPENDIX 


TABLE  I 


DENSITIES  IN  GRAMS  PER  CUBIC  CENTIMETER 


Aluminum  . 

2.67 

Alcohol  (95%)    . 

.82 

Antimony,  cast  . 

6.7 

Blood 

1.06 

Beeswax 

.96 

Carbon  di  sulphide     . 

1.29 

Bismuth,  cast 

9.8 

Chloroform 

1.5 

Brass  . 

8.4 

Copper  sulphate  solution  1,16 

Copper 

8.8    to    8.9 

Ether 

.72 

Cork    . 

.14  to      .24 

Glycerine  . 

1.27 

Galena 

7.58 

Hydrochloric  acid 

1.22 

German  silver 

8.5 

Mercury,  at  0°  C. 

13.596 

Glass,  crown 

2.5 

Milk  .  -       . 

1.03 

Glass,  flint  . 

3       to    3.5 

Nitric  acid 

1.5 

Gold    . 

19.3 

Oil  of  turpentine 

.87 

Ice 

.917 

Olive  oil     . 

.915 

Iron,  bar 

7.8 

Sulphuric  acid  (15%) 

1.10 

Iron,  cast     . 

7.2    to    7.3 

Sulphuric  acid  . 

1.8 

Ivory  . 

1.9 

Water  (4°  C.)    . 

1.000 

Lead    . 

11.3    to  11.4 

Water,  sea 

1.026 

Marble 

2.72 

Mercury,  at  0°  C. 

13.596 

GASES  AT  0°  C.  AND  76  CM. 

Platinum     . 

21.5 

PRESSURE 

Quartz 

2.65 

Silver  . 

10.4    to  10.5 

Air            ... 

.001293 

Steel    . 

7.8    to    7.9 

Carbon  dioxide 

.001977 

Sulphur,  native  . 

2.03 

Hydrogen 

.0000896 

Tin      . 

7.3 

Nitrogen 

.001256 

Zinc,  cast    . 

6.86 

Oxygen    . 

.001430 

229 

230 


APPENDIX 


TABLE    II 
DENSITY  OF  WATER  AT  VARIOUS  TEMPERATURES 


TEMPERATURE 

DENSITY 

TEMPERATURE 

DENSITY 

0°       . 

.99987 

16°    . 

.99900 

4°      . 

1.00000 

20°    . 

.99826 

8°      . 

.99989 

50°    . 

.9882 

12°      . 

.99955 

100°     . 

.9586 

TABLE  III 
RELATIVE  CONDUCTIVITIES  FOR  HEAT 

(Silver  taken  as  the  standard  of  comparison  =  100) 


Silver 

Copper 

Brass 

Zinc 

Tin 

Iron 

Lead 


Aluminum 
Brass 
Copper  . 
Glass      . 
Gold 


100 
74 
27 
20 
15 
12 
8.5 


Bismuth 
Ice  . 

Marble  . 
Water  . 
Glass  . 
Wood  . 
Air 


TABLE  IV 
COEFFICIENTS  OF  LINEAR  EXPANSION 


.000023 

.0000188 

.0000172 

.0000085 

.0000144 


Iron  and  steel 
Lead 
Platinum 
Silver     . 
Hardwood 


TABLE   V 
COEFFICIENT  OF  CUBICAL  EXPANSION 


Acetic  acid 
Alcohol  (5°  to  6°)     . 
Alcohol  (49°  to  50°) 
Ether 
Glycerine  . 
Mercury    . 


.00105  Olive  oil. 

.00105  Petroleum 

.00122  Turpentine      . 

.0015  Water  (5°  to  6°)     . 

.0005  Water  (49°  to  50°)  . 

.0001 8  Water  (99°  to  100°) 


2 

0.2 

0.15 

0.14 

0.05 

0.01 

0.007 


.000012 

.000028 

.0000088 

.000019 

.000006 


.0008 

.0009 

.0007 

.000022 

.00046 

.00076 


APPENDIX 


231 


TABLE  VI 

MELTING  POINTS 

Aluminum 

.      657°  C            Lead  . 

.      327°  C. 

Beeswax  . 

62 

Mercury 

.     -39 

Butter     . 

33 

Paraffine     . 

45  to  50 

Copper     . 

.     1084 

Platinum    . 

.     1775 

Glass 

.     1000  to  1400 

Rose's  fusible  metal 

94 

Gold 

.     1064 

Solder,  soft 

.      225 

Ice  . 

0 

Sulphur 

.      115 

Iridium  . 

.     1950 

Tin     . 

.      230 

Iron,  cast 

.     1100  to  1200 

Wax,  white 

65 

TABLE   VII 

BOILING  POINTS 

Acetic  acid 

.     117°  C. 

Water  . 

100°  C. 

Alcohol,  ethyl 

.      78.4 

Air 

.     -191 

Alcohol,  methyl 

.       66 

Ammonia    . 

.       -39 

Ether       . 

.      34.9 

Carbon  dioxide    . 

.       -78 

Mercury  . 

.     357 

Hydrogen    . 

.     -238.5 

Sulphur  . 

.    447 

Nitrogen 

.     -194.5 

Sulphuric  acid 

.    325 

Oxygen 

.     -182 

TABLE   VIII 

SPECIFIC  HEATS 

Alcohol  (0°  to  50°) 

.      .615     Iron 

.    .114 

Aluminum  (15°  to 

97°)        .     .21 

Lead     .         .         . 

.     .031 

Brass     . 

.     .094 

Marble  . 

.     .21 

Copper  . 

.     .095 

Mercury 

.     .033 

Ether    . 

.     .52 

Silver    . 

.     .056 

Glycerine 

.     .55 

Steel     . 

.     .118 

Glass   . 

.     .19 

Turpentine  . 

.     .426 

Ice 

.504 

Zinc 

.094 

232 


APPENDIX 


TABLE   IX 
LATENT  HEATS  OF  FUSION 


Beeswax 
Ice 

Lead   . 
Mercury 


Alcohol . 
Ether  . 
Mercury 


Brass 

Glass 

Granite 

Iron 

Lead 

Oak 

Steel 


Air    . 

Alcohol 

Canada  balsam 

Carbon  bisulphide 

Diamond  . 

Ether 

Glass,  crown     . 

Glass,  flint 

Glycerine . 


CALORIES 

CALORIKS 

.      97 

Silver  . 

.    21.07 

.     79.25 

Sulphur 

.      9.37 

5.37 

Tin      . 

.     14  95 

2.83 

Zinc    . 

,     28.13 

TABLE   X 

LATENT  HEATS  OF  VAPORIZATION 


CALORIES 

.     208 

.       90 

62 


Sulphur 
Turpentine 
Water    . 


CALORIES 

.     362 

.       74 

536 


TABLE   XI 
VELOCITY  OF  SOUND  IN  METERS  PER  SECOND 


3394 

4965  to  5564 

1664 

5016  to  5127 

1319  to  1368 

3287  to  3991 

4768  to  5016 


GASES  AT  0° 
Air       .... 
Carbon  dioxide    . 
Hydrogen     . 
Oxygen 


TABLE  XII 
INDICES  OF  REFRACTION 

1.000294        Ice 

1.36  Iceland  spar,  ordinary  ray  . 

1.54  Iceland  spar,  extraordinary 

1.68  ray       .... 

2.47  to  2.75  Water         . 

1.36  The  eye: 

1.53  to  1.56  Aqueous  humor 

1.58  to  1.64  Vitreous  humor 

1.47  Crystalline  lens 


332 

261 

1269 

317 


1.31 
1.65 

1.48 
1.336 

1.337 
1.339 
1.384 


APPENDIX  233 

TABLE  XIII 

ELECTRIC  RESISTANCE 

(Ohms  to  1  m.  length  and  1  sq.  mm.  cross  section.) 


Aluminum,  annealed  .  .0289 

Copper,  annealed    .  .  .0157 

Copper,  hard  .         .  .  .0150 

German  silver          .  .  .2076 

Iron,  pure        .         .  .  .0964 

Iron,  Telegraph  wire  .  .15 


Lead 196 

Manganin  .         .         .         .475 
Mercury     .         .         .         .943 
Platinum   .         .         .         .0898 
Silver,  annealed         .         .0149 
Carbon,  graphite        .      24  to  420 


TABLE  XIV 
ELECTROMOTIVE  FORCE  OF  CELLS 

These  are  only  approximate  values.     The  E.M.F.  of  cells  varies 
considerably  with  the  condition  of  the  plates  and  the  liquid. 

VOLTS  j  VOLTS 

Bunsen      ....         1.9      Grenet         ...  2 


Daniell      ....         1.07 

Edison-Lalande         .         .  .7 

Gravity     ....         1 


Grove          ....         1.9 
Leclanche  ....         1.4 


TABLE  XV 
TANGENTS  OF  ANGLES 

To  find  the  tangent  of  an  angle  not  measured  by  a  whole  number 
of  degrees,  find  first  the  tangent  of  the  integral  part  of  the  number, 
and  add  to  this  the  product  obtained  by  multiplying  the  difference 
between  this  tangent  and  the  tangent  of  the  next  whole  number  of 
degrees  by  the  decimal  part  of  the  angle.  For  example,  to  find  the 
tangent  of  38°  .7,  proceed  thus  :  — 

tan  38°  =  .781,  tan  .39°  =  .810. 

.810  — .781  =.029, 

.7  x  .029  =  .020. 

tan  38°  .7  =  .781  +  .020  =  .801. 


234 


APPENDIX 


ANGLE 

TANGENT 

ANGLE 

TANGENT 

ANGLE 

TANGENT 

ANGLE 

TANGENT 

0° 

.0000 

23° 

.424 

46° 

1.036 

69° 

2.61 

1 

.0175 

24 

.445 

47 

1.072 

70 

2.75 

2 

.0349 

25 

.466 

48 

1.111 

71 

2.90 

3 

.0524 

26 

.488 

49 

1.150 

72 

3.08 

4 

.0699 

27 

.510 

50 

1.192 

73 

3.27 

5 

.0875 

28 

.532 

51 

1.235 

74 

3.49 

6 

.1051 

29 

.554 

52 

1.280 

75 

3.73 

7 

.1228 

30 

.577 

53 

1.327 

76 

4.01 

8 

.1405 

31 

.601 

54- 

1.376 

77 

4.33 

9 

.1584 

32 

.625 

55 

1.428 

78 

4.70 

10 

.1763 

33 

.649 

56 

1.483 

79 

5.14 

11 

.194 

34 

.675 

57 

1.540 

80 

5.67 

12 

.213 

35 

.700 

58 

1.600 

81 

6.31 

13 

.231 

36 

.727 

59 

1.664 

82 

7.12 

14 

.249 

37 

.754 

60 

1.732 

83 

8.14 

15 

.268 

38 

.781 

61 

1.804 

84 

9.51 

16 

.287 

39 

.810 

62 

1.88 

85 

11.43 

17 

.306 

40 

.839 

63 

1.96 

86 

14.30 

18 

.325 

41 

.869 

64 

2.05 

87 

19.08 

19 

.344 

42 

.900 

65 

2.14 

88 

28.64 

20 

.364 

43 

.933 

66 

2.25 

89 

57.29 

21 

.384 

44 

.966 

67 

2.36 

90 

CO 

22 

.404 

45 

1.000 

68 

2.48 

TABLE  XVI 

EQUIVALENTS 

1  cm. 

=  0.3937  in. 

1  in. 

=     2.54  cm. 

1  km. 

=  0.6214  mi. 

1m. 

=      1.609km. 

1  sq.  cm. 
1  c.  cm. 

=  0.1550  sq.  in. 
=  0.0610  cu.  in. 

1  sq.  in. 
1  cu.  in. 

=      6.452  sq.  cm. 
=   16.387  ccm. 

1kg. 
11. 
1  1. 

=  2.20  Ib.  avoir. 
=  1.0567  qt.  (liquid). 
=  0.908  qt.  dry. 

1  oz.  avoir. 
1  Ib.  avoir. 

=    28.35  g. 
=  453.6  g. 

A  Brief  Course  in  General  Physics 

Experimental  and  Applied 

BY   GEORGE   A.    HOADLEY,    A.M.,   C.E. 

Professor  of  Physics  in  Swarthmore  College. 

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The   Effects  of  a   Magnetic    Field  on   Radiation.      Memoirs  by 

Faraday,  Kerr,  and  Zeeman.     Edited  by  Dr.  E.  P.  LEWIS  .        .75 

The  Laws  of  Gravitation.     Memoirs  by  Newton,  Bouguer,  and 

Cavendish.     Edited  by  Dr.  A.  S.  MACKENZIE      .         .  1  00 

The  Wave  Theory  of  Light.     Memoirs  by  Huygens,  Young,  and 

Fresnel.     Edited  by  Dr.  HENRY  CREW        .        .         .        .1.00 

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by  Joseph  Henry.     Edited  by  Dr.  J.  S.  AMES      ...        .75 

The  Discovery  of  Induced  Electric  Currents.     Vol.  II.     Memoirs 

by  Michael  Faraday.     Edited  by  Dr.  J.  S.  AMES  ...        .75 

Stereochemistry.  Memoirs  by  Pasteur,  Le  Bel,  and  Van't  Hoff, 
together  with  selections  from  later  memoirs  by  \Vislicenus 
and  others.  Edited  by  Dr.  G.  M.  RICHARDSON  .  .  .1.00 

The  Expansion  of  Gases.     Memoirs  by  Gay-Lussac  and  Regnault, 

Edited  by  Prof.  W.  W.  RANDALL 1 .00 

Radiation  and  Absorption.  Memoirs  by  Prevost,  Balfour  Stewart, 
Kirchhoff,  and  Kirchhoff  and  Bunsen.  Edited  by  Dr. 
DEWITT  B.  BRACE  , 1.00 


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Text-Books  in  Natural  History 

BY  JAMES  G.  NEEDHAM,  M.S. 
Instructor  in  Zoology,  Knox  College,  Galesburg   111. 

NEEDHAM'S  ELEMENTARY  LESSONS  IN  ZOOLOGY       .        90  cents 

A  guide  in  studying  animal  life  and  structure  in  field  and  laboratory 
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Botany  all  the  Year  Round 

A  PRACTICAL  TEXT-BOOK  FOR  SCHOOLS 

By  E.  T.  ANDREWS 

HIGH   SCHOOL,  WASHINGTON,  GA. 

Clothy  I2moy  302  pp.y  with  Ulus  tr  ations .    Price,  $1.00 


IT  is  the  aim  of  this  book  to  show  that  botany  can  be  taught 
to  good  advantage  by  means  within  the  reach  of  every  one. 

Although  adapted  for  use  in  any  secondary  school,  it  is 
designed  more  especially  for  those  which  have  no  elaborate 
and  expensive  laboratory,  but  which  have  easily  available  a 
sufficient  amount  of  botanical  material.  It  is  therefore 
particularly  suited  to  country  schools. 

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plant  life,  then  the  essential  organs  of  the  plant  are  taken  up, 
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ing plants,  in  this  way  proceeding  from  the  familiar  and  well- 
known  to  the  more  primitive  and  obscure  forms.  The  experi- 
ments described  are  simple,  requiring  only  such  appliances  as 
the  teacher  and  pupil  can  easily  devise.  Practical  questions 
are  given  at  the  end  of  each  section  with  a  view  to  bringing  out 
the  relations  more  clearly  and  to  teaching  the  pupil  to  reason 
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